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prime> n, (a1,... (al"" a1x OtZ + ... + a n 3J A • Then the thefollowing following estimate estima,te holds: holds:
+... +
pa
,an,p) and f(x) f(x) = ,an,p) = = 1,1, and =
2ff
Proof. Let y and zz run runthrough throughcomplete complete residue residue sets sets modulo modulo po-l and p, p, respecrespectively. Then the sum y11 + runsthrough throughaacomplete complete residue residue set set modulo modulo pa pO and, and, tively. + pa-l p°1 zZruns since a ~ 2 and since Rnd p> p >2,2,we we have have
f(y +
f(y) +
(mod p°).
[ch. I, §§ 4 [Ch.
Complete exponential sums Comp.fe,te sums
26
Therefore, 2 ' I(z)
pa
~
LJe
2iri 11"1
2 ' !(,+pa-t z)
,a-l P
p p ~ ~
Q
= LJ
P
LJe
2,ri 11"1
Q
P
y=1 11=1 z=1 %=1
z=1 x:l
,0-1
.
~
= LJ e
2,n
Ie,) Q
L....ti e
P
y=1
P
Cl
P . /'(11)% p ~ 211'1--
P
%=1 -
1
=P L
./(u)
27r1-
e
poe
6,,[f'(y)]. bp[f'(y)].
(70)
y=1 11=1
But then pp
,Q-t P
P
e
~P p
L 66,, pa-l Lop[f'(y)] 6,, [f'(y)] = pa-IT, p[f'(y)] [f'(y)] = p"' ,=1 ,=1
(71)
the number number of ofsolutions solutions of of the the congruence congruence where T is the
f'(y) == 0 (mod p). Since(ai, (ai,.. p) = 11 and pp isis aa prime prime> Since .... , an, > n, n, then at least least one one of of the thecoefficients coefficient,s a,,, p) is prime to p and, of the polynomial f'(y) = polynomial /'(y) = UI al + + 2a2y 2a2!1 + .,. ++ nanyn-l prime and, therefore, therefore, +... obtain the lemma assertion. assertion. T ~ n -— 1. 1. Substituting this estimate into (71), we we obtain of aa special special form. form. Let Let pp be It is is easy easy to t,o improve improve this this result result for for polynomials polynomials of b~ prime, (a,p) = = 1, and ' p0
211'i":C"
=L e ,CIl
S(a,pQ)
CIl
p
•
:c=1
Let's show that under a ~ 2 and n ~ 3 the following following equalities hold: a
(a,p ) =— S(a,pa)
{::=~S(a,pa-R)
if 2 ~ a :s.; nand (n,p) + 1. if aa ~ n +
Indeed, from (70) it follows follows that p0_I
S(a,pdl) =
e
P0 fi,,(nay"').
y=l
or
Hence under under (n,p) ==11we Hence we get get the thefirst first equality equality of (72): (72): p0_I
2wt
—
,,0_2
2iri
a(py)' PO
= 1,
(72) ( 72
,It. 4] ch. 4I, § 4J
27 21
Simplest complete sums
of p, p, which which divides dividesn.n. Then, Now let + 11 and and I'll be the greatest greatest power power of Now let aa ) n + +22 and using l)fJ + and considering considering separately sep,arately the cases + 11 ~ (p -— 1)fl using lhe the estimate estimate Qa ~ pfJ + a ~ 2{j + 2 and n + 1 ~ a ~ 2{J + 1 we obt,ain (If + + npOl-IJ-l !I"-1 z (mod pOI). (y + po-/l- 1.f)" == !1 M + Therefore, 2
S(a,j?)= y=1
z=1
pa_s_I ,.-'-1
~ = L." =
.•,n p,+1
~"".'."'" ,11" e2ft'. --., L."e
aR,
n-I
,.1
~
p5+!
,
z=I %=1
,=1
=pP+l = p,P+l
2 "
,.-'-1
2ft"1 .,.
~"",':,,"'.', L." '"e 1#
,.
£. +" I ('a"n" Up' ,', " 9,"-1)' fl·
,==1
(an,pP+') Hence" since since (an,pP+l) Hence,
= pfJ, we have =
Sea, pOt) = ,1+1
,*-,-1
E
2ft" .,"
,,8 c5,(y)
e
,=1
= pP+l
'I'"
, "-1-2,2ft',,-2wi_!!__
E
ee
,.-a = p"-IS(a,pO-R).
,,=1 y=1 Thus the assertion assertion (72) (72) is is proved proved in in full. full. TH,EOR,EM 6.6.Let THEOREM Letnnand and qq be be arbitrary arbitraiy positive int,egers integers and (a.f) (a, q) = = 1. Then Then for for the
sum Bum
— Ee 1Ft, ' 4, f,
S(a,q) = S(a,q)=>e
2 2w,,as"
.'=1 the estimate
(73) h,olds.
Proof. Since q) = = 1, then under n = 11 and Since (a, q) Wlder n estlm,ate (73) (73) follows follows from and nn == 2 the estimate Lemma 2 and Theorem Theorem 3, 3, respectively: l·espect,ively:
, 4,
,
.. =
~,"""':'", e,2.n£.-, f s=)
=0, 0,
teh'i A;' ~ Vii. e
n 6
•
•
Indeed, Indeed, under a ==11 by by (60) (60) nv'i5
n6 •
=
Let 2 ~ a ~ nn and (n,p) = p. Then Then p ~ n and, using the trivial trivial estimate, estimate, we we get
IS(a,pa)1 IS(a,pa)I Let finally 22 ~
a
0:
~
~ pa ~ pa(l-~)p~ npa(l-~).
n and (n,p) (n,p) ==1.1. Then Thenby by (72) (72) IS(a,pa)1 IS(a,p a
a(1__) . a(i__) a—I = pa(l-~) = pa-l a ~ pa(l-~) p
n. Apply Thus the estimate estimate (74) (74) is is satisfied satisfied under 11 ~ a ~ n. Apply the the induction. induction. Let Let for a certain k ~ 1. 1)n ~ aa ~ kn this estimate be valid for under 1 + + (k -— l)n We shall show show that that the estimate is valid under 11+ 1)n as welE We +kn ~ aa ~ (k ++ l)n well'. Since 1)n ~ aa — + k'n kn then using equality (72), (72), by by the that11++(k(k— - l)n - n ~ 11 + using the equality it is is plain plain that we get induction hypothesis we I
=
IS(a,pa)I
for any any aa ~ I. 1. (74) is is proved proved for Thus the estimate estimate (74) of q. q. Using Let now qq = prime factorization factorization of Using the multiplication multiplication Let now = t ..•• .• p~. be the prime formula formula (34), (34), we obt,ain obtain
pr
q
a
= [I
a c
,
(75)
v=1
where the quantities bll are are coprime coprime with with PII' ps,. We Wedetermine determinea" a,,with with the the help help of of the the where equalities (zi=1,2,...,s). a"=ab,, (v=I,2, ... ,s). (a,,,p,,) = = 11 and Then, obviously, obviously, (a",p,,) and by by (75) (75)
S(a,q) = = S(al,p~l). ,.S(a8'p~').
Alfordell's Mordell's m,.thod method
Ch. Ch. I,,, § 5] 5)
29
Hence,using usingthe the estimate e8timate (74) (74) and and observing observingthat that the the number number of primes primes less less than than nO n6 Hence, 6 we get , we get the theorem assertion: does does not not exceed nn6, Qt
IS(a,q)I~Cpt(n)pl
(1-~) .. .C,.(fl)P1(1 Q, (1-~) ... C,,(n)p.
J
~n
n' 1- 1 q n.
p) = = 11 by by (72) (72) for for any any prime p under n n>> 2, Note that under 2, qq = = p" and (a, (a,p) p"
""
L..JC
2.i"~" p" p
/
11
n. . ) . =pn-l =p =pR (1- .n
:&:=1
Therefore, in in this this case Therefore,
1I
= q1- n. S(a,q) = under fixed fixed nn and increasing the order order of of the the estimate estimate (73) (73) can can not be Thus under increasing qq the be imimproved. a1x +... + anx R , (al,'" 5(q) be a complete complete rational Let f(x) 1(x) = = alX Let and Seq) rational (al,... ,an,q) = 1land general fann form exponential sum sum of the general exponential ,q 2,..;•f(x) I(~) Seq) = q. (76)
Ec
:&:=1
In Theorem 66 the estimate 1
18(q)1 IS(q)I ~ C(n)ql-i,
(77)
6
where C(n) C(n) == nn , was proved proved for for polynomials polynomials of of the special form f(x) where I(x) = = anx R • With the help of of the the significant significant complication complication of the proof proof technique, technique, Hua Rua Loo-Keng Loo-Keng showed that that under showed under certain certainC(r&) C(n) the estimate estimate (77) (77) is valid for arbitrary complete complete well. A proof of an estimate close close to (77) can be found in in [16] [16] rational sums (77) as welL and [44]. and [441. § 5. 5. Mordell's Mordell's method metho,d Let us consider a complete exponential sum with a prime denominator P,
S(p)
= Ee
211';
At
x+ ...+an xR P
•
2:==1
proposed aa method of such such sums sums estimation estimation based on the Mordell [36] [3-6] proposed the use use of of propproperties of the the system system of ofcongruences congruences
~~. ~.: :.~.~.~. ~.~~. ~.:.~ ~.~R.}) x~
+ + ...++
x: == yi +... + ++ y:
J
(mo,d p), (modp),
(78)
Complete expon,ential exponential sums Complete
30
[Ch. [C1. I, § 5
where pp is is aa prime prime greater greater than n and the variables x1,.. where variables Xl, •••. ,tin run through through complete complete , residue set,s sets modulo p independently. residue First of First of all all we we shall shall prove prove a lemma lemma about about the thenumber number of ofsolutions solutions of of aa congruence congruence system of a more general general form.
arbitrary positive int,egers, integers, q = = LCM (Ql" be arbitrary Tk (qi,..••. ,qn) , q,,) and Tk be the number numberof ofsolutions solutions of ofthe thesystem system of ofcongruences congruences
LEMMA5.5.Let Let Ql, qi,.• •• . . ;, q" LEMMA q,,
~~..~.:".:.~.~.~ ..~.~~..~.:".."~'~'.~""~:~~"~.~'~' } , x' + + ... + tlk ... + + x% == !If +···+ +... (modqi)
x~
1
1 q. l~xlI'YII~q.
(79)
(mod q,,) (mod qn) J
Then qi
qq
qn
2
L.J e
Tk=
"'"
.fajz 12:+ +4n2:") .(4
11'1
...
91
2k 2k
II
'lfn
x1
ai=1
:1:=1
Since the product Proof. Since
equals unity, unity, if numbers numbers Xl, x1,.... . . ,,Yk satisfy the congruence system (79), (79), and vanishes equals congruence system Uk satisfy otherwise, then, then, obviously, obviously, 9
Tk= T, =
~
6'1(Xl
+... - Yk) ••• 6,.. (xi +... - Yk)'
Xl,.··",=1
Hence, using Lemma 2, we get the assertion Hence, assertion of of Lemma Lemma 5: .fa,(x1+...—yk)
q
Tk =
1
qi . . . q,, al....,an Z1,...,ykrl
~
1
e"
L.Je
qj
21ri(.,t z + + ,,:a:") A
o ••
1
2k
9"
r=1 x=1
In particular, under kk = and qlq1==... = n and ... = qn = pP ·itit follows follows from from Lemma' 5, 5, that that 1
i'
2w,
(80)
where Tn is the the number nwnberof ofsolutions solutions of of the the system system(78). (78). LEMMA LEMMA 6. Under any n n ~ 1 and a prime prime pp>>n,n,the thenumber numberof ofsolutions solutions of the the system syst,em (78) satisfies satisfies inequality in,equality T" ~ n! ph. p".
31
MordeJi's method MordeJl's method
5) Ch. I, §§ 5]
. . ,An) , A,, be , A,,) be be the be fixed fixed integers, 0 ~ A" ~ p -— 1, and let proof. Let A1,.. AI, .... ,An let T(A1, T(A I , .••• number of solutions of the the system system of ofcongruences congruences
~~,,~.:".:.~.~.~. . ~. ~.1})
}
xf +··.+ x: == An
(modp) (mod p)
1
~ ~ p. x"
(81)
We shall show show that that We shall (82) T(AI, ... ,An) ~ n!. (82) Indeed, we intro,duce introduce the notation for for the elementary symmetric the following following not,ation symmetric functions functions and the sums sums of powers powersof ofquantities quantitiesXl, xi,.•••.. ,X , and n:
0'1
= Xl +... + 5,,,... 51+ ... +X ii..•• .IX5,,, n, n , ••• ,0'" = Xl
81
+ ... + ... ,sn=x~+I + xn• =Xl+,".+Xn, .. +x:" = .
Let Xl, arbitrary solution solution of of the the system system (81). (81). Then, Then, obviously, obviously, Let ii,. •.•• ,Xxnn be an arbitrary (modp), 81 == At, "'" 8 n == An (mod p),
and using the the Newton Newton recurrence recurrence formula formula
=
110.,, vO'" = 810'''-1 -— 820'''-2
+..."' T 8,,-10'1 ± s", +. J
under = 1,2, 1,2,... under vji = ... ,n ,nwe we have have A,, (mod p). +... (83) + ... =f A,,-lO'I ± A" Since Since pp isis aa prime prime greater greater than n, n, then then (v, (lI,p) p) ==11 and andthe thecongruence congruence (83) (83) is is soluble soluble 110'"
==
A10'1I-l -— A20'1I-2
for 0'". oP,,.FraIn From (83) (83) we we get successively for B'uceessively ,0.,, O"l=""l"",O'n=J1.n
(0 p,, p — 1), (mod p) . (O~JLII~p-l), (modp)
where the the values valuesPI,. ni,. .. ,ltn uniquely by bysetting setting quantities quantities AI, A1,..., where are determined uniquely ... ,A,,. Ani , with one one of of the the permut,ations permutations of But then every every solution of the the system system (81) (81) coincides coincides with of the roots of the the congruence congruence f
•
x
n
fl
I-'lX -
-
1
+... ± Itn == 0
(modp) (mod p)
with fixed coefficients cOlefHcients and, therefore,
T(Al" · . ,An) ~ n!. Now, since P
T,, TR
= =
L
+ V:), +... + + ... + + tin,· y,,,... yf +
T(y1 T('II1 +...
f ' ,,
y1 ,... =1 111 "",'n=l
we get get the lemma assertion: we Tn ~
'P
L
'1 ..... ,,,=1
n!=n!p". n! = n! p•.
f.
I
(ch. I, § 5 (Ch.
Complete exponential sums Complete
32
immediately, that that under any Not,e. the equality equality (80) (80) it it follows follows immediately, any Note. From this lemma and the n ~ 1 and a prime prime p> p >nnthe thefollowing following estimate estimat,e holds: holds:: P P
P P
2:
2: e
Gl1 ••• ,I1.=1
TUEOREM atX
7. Let n
~
2n
alz+...+anr" 2 ' tltx+ ...+an z " 2n p
lI'l
= pllTII ~ n! p211 .
:1:=1
greater than n, (ai, . . ,an,p) = 1 and and f(x) f(x) = = 2, p be a prime great,er (at, ....
+... + anx n • Then
p I'
2 '•f(x) /(,;)
2: e "'1-,-
1 ~ npl-n.
,;=1
Proof. At first we shall consider consider the the case case (an, (as, p) p) == 1. 1. Let Let integers integers A ,\ and J.t vary in the bounds bounds 11 ~ ,\A ~ p - 1, 11 ~ Jl ~ p. Arrange Arrange the the polynomial polynomial f(Ax + + It) in the ascending order of of powers powers of x
+... +
= bo(A,/1) + b1(A,
f(Ax +
p)z"
(84)
observe that that and observe bn -
and and
= aMA"
1 ('\,
= (nanJ.t + J.t) = an-l)'\ n-l. + an....i
D,enote number of of solutions solutions of the system system Denote the number bi(A,1i)
p)
(85)
b,, , n ). is plain, plain, that that H(b1, H(51,. . . ,b It is . . ,,bn ) does by H(b1, ... ... does not not exceed exceed the the number nmnber of solutions of the system made up up of of the the last last two two congruences congruences of the system system (85): (85):
+
1
aDA"
j
b,
(mod p),
since (nan,p) (nan,p) = and, therefore, therefore, since =11 and and (A,p) ('\,p) = =1,1, (86)
By (25) for complete sums the equality . f(x) /(z) 2n 2n p '"" 211'1LJe P :1:=1
p
=
. f(Ax+p) I(~:I:+") 2n 2n
E21r* LJe
'""
:1:'=1
2'1f1
,
3,3 33
AAordell's Morddll's m,ethod method
Ch. I,4 §§ 5]
Hence by by (84) (84) after after the summation with respect respect to A we have holds. A and JI"p, we holds.. Hence p
f(x) 2t*
2ir; —
P—i
=
>2
(p
2n Pp . I(>':z;+~) f(Ax+p) 2ft 2,rs '"" 271" L..Je P
P
>2
p1 r=i :&:=1
A=1
P' p
2n p P _.. 61(>',I')Z+ ... +6.. ('\.~)a:R 2ft bi(A,p)x+...+b,,(A,p)x" '"" 2,...
L..Je
>2>2
P
x1
A=lp=1
:1:=1
ii), ... , n (A>Il) Grouping summandswith withfixed fixedvalues valuesb1b1(A, (A,I-&), ,b Grouping the summands /1) and and using using the the esti. mate (86), we get .
.
2n
p
2n
>2
p
z=i
x1 2n
p . >2e27rt
P
(
Hence by the note note of of Lemma Lemma 66 we we obtain obt,ain the the theorem theorem assertion assertion for for the the case case
(an,p) = = 1:
f(x) 2n ./(z) 2n
p
Ee
211"1-
~ ~
p
:.:==1
Ee 11'1-,p
2 ' I(z)
n n'p2n < n 2 n p 2 n-2 (p -1) , p(p—i) ·
P
1
< npl-n.
x=1
Now we show show that that the general Now we general case (a1,. (al'.'.', an,p) == i1can can be be reduced reduced to to the the case case when leading coefficient p. coefficient of the polynomial is prime to p. Indeed, let (a.,p) = 1 and a s +l == ... == an == 0 (mod p), 1 ~ s ~ n. Thenwe Then we obtain obt,ain . .
pP
~
L..JeS
2 ' f(z) f(z)
p ~ 271'i 2,rs
-71'1-
= = L..J e
P
z==l
atx+...+aax 41:&:+ ••• +a,x' P
1
1——
1
1——
x=l
The theorem is proved in i.n full. full.
Not·e. AAsubstantial subst,antialimprovement improvementofofMordell's Mordell'sestimate estimatewas wasobtained obt,ainedby byA. A.Weil Weil[48], [48], Note. who showed,that thatunder underprime primepp> (an,.... . ,, an, p) = who showed, > nn and (an, =11 the the estimate estimate .
p
is is valid.
Complete exponential sums Comp,/ete
34
(ch. I,I, §§ 6 (Ch.
§ 6. Systems of of congruences congruences
is the use of the estimation for One of the main main points points of ofMordell's Mordell's method method (§ 5) is the number number of of solutions solutioll8 of of the the congruence congruence system ) ~1.~ ~'.~ ~~.'~ ~~ ~ ~.~.n.} n_ni ft :
.... :
X11...VXn.Y1r...TYn xi +···+ x: == yf +···+ y:
(modp),
fl
L
I
where pp is is a prime Hereafter, congruences congruencesofofthe the same same form form but where prime greater greater than than ,z. n. Hereafter, with respect to to distinct distinctmoduli moduli being bei.ng equal equal to togrowing growing powers powers of a prime p will will be be great importance. import,anee. For For the thefirst first time timesuch suchsystems systemsof ofcongruences congruences were applied by of great Yu. V. V. Linnik [34] for the the estimation of Weyl's by Vinogradov's Vinogradov's method. Yu. [34] for Weyl's sums by n(n4+i) 1 ~ 1, 1, kIe ?~ n("4+ ), p be a prime prime great,er be greater than than n, n, and let let Tk(pn) Tk(p') be tbe of solutions of the system syst,em of congruences the number of congruences
LEMMA LEMMA7.7. Let Let n
(modp)
Then
)
n
(87)
n(n+1) 2
Tk(p'2)
Proof. Under Undernn==1 1the thelemma lemmaassertion assertionisisevident, evident,so soititsuffices suffices to consider the case n ~ 2. Take Take qlqi=p,...,qn=p" = p, ... ,qn = pnin inLemma Lemma5. o. Then we obt,ain obtain
"
n(n+1) 2
ajl
P'
e
I
x=1
of the the summation summationover overat, al,... We split up the domain domain of ... ,a'n , into two parts: part,s: p't
p
aj'tl where the summation in where
a1,...,a't
2:1 is extended over over n-tuples n-tuples a1 at .... an sa~i,sfy an which salisfy . .
p\a2,p
\a3,...,p
and for for 2:2 n-tuples are taken t,aken into account, for for which which at least for for one one of of E2 only those n-tuple8 ,i in the interval IJ interval 22 ~ v1J ~ n p"-l is not a divisor divisor of all.
__________ ___________ Ch. ch. 1,I,
S6]
Systems o(cosgruenccs
35
case,determini,ng determiningb1 b1,. In the first flrs't case, t • • •. t,b,, 6ft with the help of the t,he equalities equalities
..., a,,
a1 = b1, a2 = b2p,
we get
2irs (!-1!+ ~.".' '., '. .211'i---+ LJe .. "
( . 811:
II
4
=
. ''.' " 2wi "1~ =~ LJe
Sa )..
" P"
pU +...!L;, ."
z=1
.I.
+
.I.
"
II+.,,,a
•••
.=1
,P
~.' . 211'" =Pn-l LJe
.=1
61.+···'+...." . ,P ' ·
Therefore
f:
_~z +...+ -;:") P p"
2iri ( e21l'i
2k
2k
=
p2R1t-2ft
~-1
t
p
e 21I'i
'l.+..~H.. z..
2k
211
.=1
bix+...+&,,x" 2n
P
and using using the note of obtain and of Lemma Lemma 6, 6. we obt,ain
.n). 2'
2 " . .!!.=.+... +~ . . ~.".""" ..... Le .. , ,n
PD p.........
(....
. ...
2k
.
x=1 al,···,4" 11:=1 P
P
(88)
P
b1,...,b,,=1 x=1
In the second v in the second case, case, there exists exists an integer integer" the interval interval 22
v ~ ~ II
p.-l ~ a"pM-II. Therefore" Therefore, ,,-1. . . . ( alP ,G2P. ..-2 •...
where 2
~
") ,a,,,p") ,G,.,P .= = PR-a ,
n. But a ~ R.. But then
Q
a-I (lIP
= ClIP"-a , • • • ,tit. = VaP.a-a , 1..
(bi,...,b,,,p)=1, (b'l' .... ,b., p) = 1,
~
and using Lemma Lem,ma 4, 4, we we get ~ LJe '..••..."
..
6)
•
. ..........•... 61ft
.:==1
= =
(·.·. . •
a.• +
'
f: z=1 .:==1
.•••
+., .".").
2wi ehi
,It bix+...+b,,z .t.+.;:......
= p,.-a
' Ee ,"
z=1 .=1
211'"
'11:+ ... +6,,11"
,. P"
that nn such such tha.t
[Ch. (rh. I,I, §§ 6
Complete exponential sums Complete
3,6 36
n(n+1) it follows that Hence, since k ~ n(~+l), follows that pn "
>22 >2e Le L z1
anx") ,n
2iri. (al~ -+...+ P
211'1
2
41 ••••• 4n
2k 2k
~=1
n(n+1) —
observing that that under n Now, observing
—
(n
2
(89)
~ 2
l)2k_1 > 2k -— (n — 1)21c ~ 2k(n -— 1)2k-l nfl2k - l)n-l ~ n! (n _1)21: n(n — > n(n
from (88) and (89) (89) we we obtain the the lemma lemma assertion: assertion: n(n+1) 2 Tk(pn) =p---2-
Ll + L2 >22)
n(n+l) (
)
(
Al J•• 'J4 n
~n!p
41 J' •• ' . "
n(n+1) 2nk2nk— n(n+l)
n(n-4-1) 2nk2nk— n(n+l)
+(n_l)2k +(n—1) pp
2
2
n(n+l) n(n+1)
2nk— 2k 2 nk--2 ... >2 >2 e
,"
L'" L Le
&1==1
=
""" L...J e
a,,=1 4 n =1 x=1 x=1
n(n+1) n(n+l), 2k p - - 2 2 -
mp" m pn .2
4,,:z1
x=1 x=1
n
0
)
2k 2k
____________________ Sys:tems of ofcongruences congfuen,ces Systems
Ch. i,I, § 6] 6] ch.
37 37
Hence, since by Lemma 77 Hence, 2 k—
Tk(p")
2
we obtain the estimate estimate (90). (90). we . ,, X n run Let L.J"'1,. ~!Q •• , . denote the sum, sum, in in which which the the summation summation variables variables x1,. Xl, ••• through complete complete residue sets set,s modulo modulo pa pO' and and belong belong to to different different classes classes modulo p. .
= a1x 1(x) I(x) = ala: + + anx n , + ... +
LEMMA8. 8.Let Letppbe beaaprime primegrea,ter greaterthan than n, n, aa ~ 2, and LEMMA
with the help of the equality Let Sa( aI, . .., . ,,an) be defined defined with e,quality Let Sa(ai,. ,ol 2m /(%1)+ •.• +/(x,.) 2ir, 5'a(al,'" ,an)
L
=
,0
e
Then a ) -S eN (a 1,···,n
{
P(a-i)ns1 (b1 , • • ., bn )
if a"=pa-1b,, otherwise. otherwise,
o0
(v=1,2,...,n), (v=1,2, ... ,n),
Proof. Let Let us us change change the the variables variables X"
(v=1,2,...,n). (v= 1,2, ... ,n).
= YII + P0'-1 Z.,
different classes, classes, then then the Since Sinceby bythe theassumption assumptionthe the quantities quantities Xl, xj,.••• ,X , x,,n belong to different belong to to different classesmodulo moduloppas as well. well. Therefore, Therefore, using using quantities Yl,' different classes quantities Yi,.... ,Yn , yn belong that (mod pa), + f(yv we obtain .
.
Sa(ai,. .. p
e
=
>2
n =p
e
%1 ••••• %n=1
111,···.lIn
,.-1 P
2"'; /(1/1)+ ...
L
2
e
+/('n) 0, [f'(Yl)] ... 0p[/'(lIn)].
pO
(91)
lit •• ... 11,.
and (at, (ai,...... ,, an,p) Sincefl(y) f'(y) = at a1+2a2Y+ +2a2y+.. Since ... +nany"-l, then under prime p > n and p) = = the congruence congruence f'('II) f'(y) == 00 (mod p) can values of 11y 11 the can be be satisfied satisfied by at at most most nn— - 11 values from different different classes classesmodulo modulop.p. In In the the sum (91) (91) the quantities Yl,.,., Yn from belong to y, belong different different classes classes and, and, therefore, . .
1
.8
=
{
0
((al,. )= , an,p) at, . ,an,p = p, (as,... ,an,p) if (at, ,an,p) = =1.1. 'f lif
.
[Ch. I, § 6 [Ch.
Complete exponential expon,ential sums
,a,,) ,an,p) But then then by by (91) (91) the sum S'UDl Sa(ai,. Sa-Cal"~ ... ,a under (a1,... (al," .,an,p) n ) vanishes under (ai,... (al , ... ,, a,,,p) an, p) = = p, then .
pa_I pClr-t
= 1.1.
If
2wi
E
Sa(ai,...,an)=p Bo(al"" ,an) = pR >2
e
't,··"Y" Thus
.. ,anp-l) f ... -— {pnSa_l(alP-l, S o (at,·· · , an ) 0 —
if (a1,... (at, .. . ,an,p) =P 'f (a1,... (aI," .,a,.,p) = 1if = 11..
0
1
92 (92)
Applyingthe theequality equality(92) (92) ,anp') we get Applying to to 5 0 Sa_i(aip',. - 1(alp-I, ... . ,anP-I) .
S (a a)Sa(ai,...,an) 1, • • • ,n Q
{
- 2 , ... ,anP- 2 ) ' f (al,···,a n ,p2) =P, 2 P2n S 0-2 (alP 1if (aI,...,an,p2)=p2, 00
otherwise. otherwise.
Continue st,ep we we obtain obtain the thelemma lemma assertion: assertion: Continue this this process. process. Then after a-I — 1 step . Sa-(al"'" an) = = fp(a_1)nSi(aip_(a_1),...,anp_(a_1)) l)n S1 (atp-(a-l), ... , anP-(a-l») .
{p(a1
=pO_l, if (al"'" an,pa-l) = pQ-l, otherwise.
O 0
LEMMA9.9.(Linnik's (Linnik's lemma). lemma). Let A1,. LEMMA At, ..... ,,A,, An be fixed fixed int,egers, integers, pp aa prim,e prime greater greater than T*(A1,. . . n and let T'*(Al, .. " , As,) be the number An) be numbcl' of of solutions of the tl1e system system of of congruences congruences
~1.~ '.~~~.~ ~~ '«~n'lOo.d .Pp'>R) } , (modp)
:'.
•
xi +···+ x: == An
(modp")
vaJ.·ia,bles run through complete complet,e residue sets sets modulo pn and belong belong to t,o variables different class,cg modulo p. Then different classes mod ulo p.
where the
n(n—1) n(n-l)
··T*(,xt, . .. ,A n ) ~ n!p-2n!p 2 n - 2 x 2 + ... + anx Proof. Let 1(x) = alpn-l x + a2p a,,x"n and according to the notation Let lex) according to a2p"2x2 of Lemma 8 •
2ir,
>2
e
~l, .... ;l:n
Using Lemma 2 we we obt,ain obtain Using pfl p"
n
~ll ••• ,Xn
v=1
L II 6pv(x~ + ... + x~ >2 pfl p"
n(n+l) n(n+1) 2 T*(A1,.... . . p-2-T'*(Al' p
,,=1
,An) = =
,
>
at,... ,4"
L,:I:"
>.
Xlt •••
2' (
2 11'1•
e
41
'
Xl
Av ),
+...+2:" -..\1 ++ ••• :1:;.'+•.. +:1:: -..\. ) ..• +4" + P p" 1,,
_____________ ________ 39 39
Systems o(congn,ences Sys,tems of ~
Cit. I, § 6] '] ch.
.. ,I aM is wbere the summation summation with with respect respect to to a11. 61 I ••• i,8 extended extended over over the the domain domain where 1 ~ (11 E:; P, . · · ,1 ~ a.. ~ pM.. Henceobservingthat Hence observing that
E .".•. .-.
.' .,(. . .1+.··+·. .. .:+,.+.:). , +...+..
e21F'·1
Sl ••• ·,·" -
2
= =
pfl
e
Sit···.·..
we have a T*('\l' , ... An) . . I
.L
A ?' -2 __ A" ).. . —2 •(.. . 41 l + +.... ~ S (.. ..-1 \. 1ft k , 00. ,. P" ••. = P L . . , ' ... L..,,,alP , • ·..· I a'Rr'
n(R+1) ,.(R.+l)
.pp .
- -2 - ~ 2
'.........
ajl
at=1
.'
a.=l
Determi.ne t,he ... ,b. with the help help of of the the equalities equalities Determine thequantit,ies quantities~,...... (11 = bl , ai=bi,
"2
,a" =
= pb21.'"
p,,-lb._
Acoordi,ng to Lemma 8 According
S.(alp·-l, ... ,a.) .
,b,,) = {. pfl(tI-. l)Sl (b'I' _. - ,b,.)
(v == 1,2, 1,2,... =p"1b,, (II =p.,-lb" ... ,n), otherwise, otherwi.se, if
o U
I,
a,, 0"
, E
where
P
-
Therefore, n(rt—3)
p
2
T(A1,.. .,A,,) p
=
Si(bi,. ..
, L
p
p
=
2W,
P
-
Now, using Lemma 2, we obtain n(n—1) nCn-l)
T*(Ai,...,A,,)=p 2 T:*(;\l, · · · , ~ ..) p-2-
..ii
,p
L II 6"(%1 +·... + a:: - A.,)
=
.,=1 tj,...,Xn.. v=1
.1,.0.,~
,p
3:1
nii
L E I16,(:,;r +... + %: - A,,),
"'0,... =1
,.,=:;:1
_(ft-l) n(n—1) 2 = p-2-T(Al' . _. .. ,Aft),
•
[Ch. I, §§ 6
Complete exponential sums Comp,/e't••xpan,ential sums
40
where T(Al"'" nwnber of of solutions of the system of of congruences . . , An) is the number
1 ~ x" ~ p. 72i
I
n! by (82), the . . ,, An) ~ n! Hence, because T(Al" .. the lemma lemma assertion assertionfollows: follows: n(n—i) n(n-l) 2 T*(Ai,. T*(Al"'" An) ~ n!pn(n-l)/2. . .. An) ~ p-2-T(Al"'" .
COROLLARY. Let of solutions of the system of of congruences COROLLARY. Let T:(mpn) be the number of
x1 +
—
0
(modp)
1~
i
(modp)
Xj,Yj ~
mpn,
i- j => Xi '¢ Xj,
Then
Yi ¢.
Yj Y, (modp).
,z(n+i)
2 —
2
(93)
. . ,, Yn runs runs through a complete Proof. Since Sinceeach eachvariable variable among among x1,. Xl, ••• complete residue system modulo p" m m times times (under (under the the additional additional conditions conditions i f; jj => Xi ¢. Xj, Yi tem modulo pR y1 ~ y, (modp)), then using the lemma we obtain Yj (modp», then using the lemma we T:(mpn) = m 2nT:(pn) ph
=m
L
2n
+···+Yn, .. o,yj+ ... +y:)
T:*(YI
>
1I1,.. ·,Yn
n(n—1)
p
2
2 2_ n(n+1) 2 = n! m2tlp
Sums with exponential function §§ 7. Sums Let a be an Let an integer, integer, in m form
2 ~ 2
and q
~ 2
be coprime positive int,egers. integers. Sums of the coprime positive P
aqr:
S(P) = ""'" LJe 211'im :.:=1
rational exponential sums cont,aining containing an an exponential exponential function. function. In the invesare called ra,tional invessome properties properties of of the the order order of qq for for modulus~ modulus m. tigation of such sums we shall need some
Sums Sums with exponential ~ponent;al function function
7] Ch. ch. I, §§ 7)
Let p be a prime, prime, m
41
==pm1, pm!, rr and andr1rl be bethe theorders orders of ofqqfor for moduli moduli m m and and m1, ml,
respectively. We shall shall show, show,that that ifif rr respectively. We
~ r1 rl and
p\ml, then the equality equality
rr=pr1 =pTl
(94)
holds. (mod m) we 1 Indeed, since since ml mi\m, 1 (mod Indeed, \m,then thenfrom fromthe thecongruence congruenceqT qT == 1 we get get qT qT == 1 (mod ml) and, \r. On (mod and, therefore, therefore, r1 71 \r. Onthe theother otherhand, hand,frQm fromthe thecongruence congruenceqTi qT l == 11 + uimi, is a multiple of of (mod ml) mi) we obtain = 11 + (mod obt,ain qTl qTt = Ulml, where where u1 UI is an integer and m1 ml is assumption. But then p by the assumption. But p qPfl qP T l
= (1 + Ulml)P == 1
(modm) (mod m)
:F rr and and pp is is aa prime, prime, the equal~ty (94) follows: follows: 1 rrj'\p, TT1 \p, T1"l-1 = p, r=pr1. r = PTlrrr'=p, the prime factorizationof ofm, m,Tr and r1 now ~ rn be be o,dd, odd, m m == pfl ... P~' be the Let now prime factorization 71 be
and r\]JTlr\ pr1. Since \r, Ti Since r1 71 \T, Tl
. . .
orders of of qq for for moduli moduli m m and PI pi .... .-P"~ respectively_ respectively. the orders We determine the quantities flu,.. We det,ermine PI, .... ,,P. with the help of the conditions conditions _ qT' qTl —
11 --=
U '0 p{Jl . . . pP,', 1 .•• 8
(u p1··· 1· (u0, 0, P1 -- 1. .. . p , ) =
(95)
For definiteness definitenesswe wesuppose supposethat that in in the prime factoriz,ation factorization of of m in those primes, which which For satisfy the the inequality inequality a" a,, > so a" a,, > /9,, satisfy > f3,,, {3", are put at the the first first rr places places (0 (0 ~ rr ~ .s), s), so P" ur.der IJz' > r, r. Further let under v1/ ~ rr and a,, a" ~ /9,, p" uI~der _ (JI m1 -=p1 mI PI
(Jr
.. 'Pr
. .
Qr+l Pr+l •.
a, a
·P. ·
From the the definition of Tlr1 and and the equality (95) it follows that the order From definition of equality (95) follows that order of of qq for for modulus ml m1 isis equal equal to to 71 r1 and ~odulu8 qri + u1m1, qTt = 1 + Ulml,
(uj, Pu (Ul' PI ... .pr) . Pr) ==1. 1. .
Let us show the validity validity of the equalities equalities
m = + (u, PI .. . Pr) = 1 and T = - T l . (u,pi...pr)=1 ml Indeed, let m2 = pm1, where p is any number among the primes Pu,... Indeed, m2 = pml, where p is any number among the primes PI , ... ,Pr.
qT=1+um, qT 1 urn,
(96) Let 'T2 r2
denote the order denote order of of qq for for modulus modulusm2. m2' Obviously ObviouslyT2 T2 ~ r1 7"1 (for (for otherwise otherwise we we would would 1 have m2\qT' pm1\Ulml, \uimi, which m2\qr — -11 and pml which contradicts the condition (uj, (UI' p1 PI .. ..p,.) Pr) == 1). 1). . Since, besidesthat, that, p\ml, p\rn1, then by (94) But then Since, besides (94) r2 72 = pTI. PTI. But .
qP7i1 ~ = (1 (1 + + Ulml)' q,,'r qT2
qT2
where u2 'U2
+ u1m2 (modp1 ... .. Prm2), == 1 + Ulm2 (mo,dpl .
= 11 ++u2m2, = U2m2,
u1 (mod pi == Ul PI .. . p,.), Pr), and, a.nd, therefore, (u2, (U2' Pj PI .. . Pr) Pr) = = 1.1. Thus Thus . . .
T
qq
2
= 1 fU2m2, +u2m2, = 1
.
m2 ml
2 (u2, Pl Pu.. = —Ti. (U2' .. ..Pr) Pr) = 11 and T2 T2 = m 7I •
Repeating this being equal equal to to each each P" p', (IJ (ii = 1, 2,. , this process processa,, all— - /9,, P" times with pp being 2, .... ,r), obtain the we obt,ain the equality equality (96). (96). .
(Ch. I,I, [Ch.
Complete exponential sums
42
117
2, pp be a prime, 2, prime" m = pmj, pml, rT and andr1 Tl be be the theorders orders of of qq for for and p2\m, then under moduli mt, respectively. respe,ctive1y. If Tr 1 Tl &l1d undm- any a not not divisible divisible moduli m m and andm1, by p byp
T'HEOR,EM THEOREM8.8. Let Let m ~
7r
. at/' —
2w, '""'. . . . . . . . . . . 2wi - m
L..."e
= O'. ,
•
(97)
1:=1
Proof. Let LetTTdenote denotethe thenumber numberoforsolutions solutionsof ofthe t,hecongruence congruence
== fl"
qUsi,n~ Using
(modm), (mod m),
Lemma 2, we ~ obt,ain obtain
I
7
— qV) =
T=
In
7
2w;
I
In
m a=I z=1
T. But then On the other hand, hand, obviously, obvi,ously, TT = T. In
L '"
2
~ 271 2tri -(' L..." e '"
mT = = mr. = mT mT".
a=I 4,:=1 x=1 ~=1
(98)
Therefore, by (94) 2
(e,p)=I In
=
E s1
= mT mr =
r
in 1ft
Ee27tmn
—
L a1 (CI,,)=, 4=1
USj "'1
,Ta
E Le
. Cll'. 22 21ft -;;-
gjI z=I .:=1
r .a.L..." e ,. '""' . . . . •.:. . . . . 2wi -.-
2
1M
.,=1 x=I
= mT = mT -
—
=
p'Jml TI = 0. O.
4t=1
Hence for for any any CIa not divisible by pp we we obtain obtain the theorem assertion Hence divisible by r
T
.,.
" ,. . . . ~hr' --L..."e ..
E
=0. = o.
1:=1
Let us show that ifif at at least 1e'astone one of of the the theorem theorem conditions conditions and
p2\m
Sums expon,ential function function Sums with exponential
Ch. ch. I, §§ 7]
43 43
satisfied, then the sum is sum (97) (97) might be be not not equal equal to to zero. zero. is not satisfied, Indeed, > 22 be be an an arbitrary arbitrary prime, prime, m and qq be be aa primitive primitive (a, m)=l, m)=1, and Indeed, let let pp> m = 2p, (a, root of 2p. we have 2p. Then we rr
49#1:
2,ri— 'Jr. -;;;;m
" ' " e2 L...i
r". =
x=l
211'1' _4(_2_x-_l_>
P
L
e
"i
2,
= -e =—e
a 211'1 2
2=1. = 1.
x=1 z~p+l 2
r1 is obviously satisfied, but In this example but p2 p2 ~2p, and the the second second example the condition 7r ~ 71 condition is violated. Let now primitive root of p2, = pP — - 11 and, p2, and and q = gP. Then rT = Let now m m = p2, p2, 9g be a primitive using using the the equality (98), we obt,ain obtain
p—i 2ir1—
p2
p(p-l) (a,p)=1
p1 2,ri — L L e ,2 1) pp22
=
1
p-l 211'1 ag"· 2
= pep ( - 1) a=i 4=1 x=1 :1:=1
(a,p)=1
Le
%=1
,
"'"
L...i e ( — 1) aj=i x=1 :1:=1
—
r
max
2 ajgW 2 .. al'z
1'-1 1
e
2tri
211"1--
P
— 1, = pP -1,
aq21
m ~.;p:::l.
In this case the condition condition p2\m p2\m was was fulfilled, fulfilled, but 7 = 1"1. conditions, under under which which the the complete completesums sums S( S(r) Another form of conditions, T) vanish, is shown in the following following theorem.
pr
.. . P~' be THEoREM9.9. Let Let m m == 1 ••• be prime prime factoriza.tion factorizationof ofodd oddm, m,7 r be be the the order of THEOREM for modulus modulus m m and and the quantities /3k,... determined by by the the equality equality (95). (95). If If q for PI, ... ,,{jIJ be determined there exists exists vi/ such such that that Q II > PI! and aa¢.O0 (mod p~l1-fJP), then there
rr
a9~
"'" L...i e211'1m :a:
Proof. Chose Proof. Chose that value value
II,
alJ
=0. = O.
=1
which satisfies satisfies the conditions
> {ill'
a
=
and write a in the form a = = pap -fJp -"'( at, where '1 ~ 11 and Let a1, where and (a', (a', p,,) p,,) = 1.1. Let m == P~" -p" -'Y m', rn' r' and for moduli moduli m' m' and m", m m' = puinU, pI/mil, 7' and i-" T" be be the the orders orders of q for respectively. Since p~., \m, then respectively. then pe l1 +"'t\m'. But then pe., \m" and and by by (96) (96)
T,, =
m" T1, mi
m'
m'
mt
mil
,, = —Ti m' m'" T = -". —r ,, =p,,7". PvT 7 = -71 = mj rn"
Complete exponential sums Comp,/ett
44
[Ch. I, § 17 [Ch.
divisibilityofofm'm'bybype,,+'Y p' it it follows follows also, also, that p~ \m'. Thus From the divisibility
r' t= r",
in' =p"m", = p,,rd', m'
and
TI
(a',pp) =
1.
Therefore, by Theorem 8 T'
Le
.
•
a't/'
21r'-·
m'
=0.
~=1
Since qT in') and in) and m'\m, then Since qr == 11 (mod (modm) then qT qT == 1 (mod (modm') andT1 r' is a divisor divisor of of T. T. Now, using the property property (26), (26), we we obtain obt,ain the the theorem theorem assertion: assertion: r 2,ri. '" L...Je m = L...Je T
('
T
•
' " 211';.!!........
a' ,:II
211'1 - ,
m
.,.1.
= T"
~=1
x=1
a' ,"
r ' " 211"1-,
L...Je
m
= O.
x=1
that the in Note that theTheorem Theorem99requirements requirement,s can can be be relaxed, relaxed, namely, namely, the condition of m being odd may suffices may be h,e omitted. omitted.InInorder ordertotoprove provethat thatit it suffices(see (see[321) [32]) in determining the quantities the. PI, .... . ,,P. equality fl8 to use the equality 1 -- u 0p(Jt 1
q(IL+l)11 q(l&+l)Tl _
pfJ,
" •• ,
where IJ = 1, if m 1 (mod (mod2), in == 0 (mod 2), r1 ;1 == 1 2), q == 33 (mod4), and It = = 00 otherwise, instead instea.d of the the equality equality (95). (95).
THEOREM 10. Let Let m ~ 2 be in) = TUEOREM 10. be an arbitrary arbitrary integer, int,eger, (a, (a,m) 1, (q,m) = 1, (q, m) == i, 1, , and Tr be modulus m. m. Then the order of of q for modulus Then the the estimate estimat,e r
T
Le
(s,. mT
.(aq' +bZ)
2 1r' •
()
~.;m
z=l
holds under any any integer integer b. b.
parts Proof. Since Since the fractional part,s
{a::} m
and and
{b:} T
have the same by (28) same period period T, r, then by (28) the sum (99) (99) is aa complete complete exponential exponential sum. sum. But then under under any any integer integer z
r Le r
2'
11'.
r 2 ' (492+.1: + 6l1:+6Z) bz\ bx+bz\ (49.:6 + 6X) m T = Le 1r. ---m= T
-T-
:1:=1
te x=1
:1:=1 21ri
(ll:; +":)
=
=
te
z=1
. 21ri
(Il':" +":)
,
Distribution of ofdigits di6its in in complete comp./ete period period of 0.(periodic p.riodic fractions fra·ctions
8J Ch. ch. I,I, §§ 8)
Therefore,
~ 211'i ". L...Je
(ata: +6:t) r
m
,211'i
=E
r
~~1
r Ee
r
2
(4f~, r + 6~ )
45
2
r
m
%=1 :.:=1
rr
~
L-Je
z1
2 ' #.
faqzz bz\ 2 (4',##%+":1:) 2 -' m
T
:1:=1
because the congruence Hence the theorem assertion assertion follows, follows, because (modm),
q:t=q"
1 ~ x,y
~ T,
=
satisfied for for xx = y only: is satisfied 2 1~ Ee (8""iiI +6:1:) r ~ - L...J r
9':
2 2iri' 11'1
~ 211'1 L-Je
(a,Zm
%
+6:t) 2 T
z=1 x=1 x1
x=1
T %=1
1
r
T
b(z—y) m "b,(x-,> m
,
. (q:D -q~)z
211'1 -, - ~ 21r1 =— = - ~ L-J e T L...Je
=m =:;:
m m
%=1
T x.,=1
b(x—y) ~ 211'; 2wi "(x-,) r On1 ( qX L...J ee T T
)
-—
q11 q11)
= m, in, =
x%.11·=1 ,y= 1
+hX) Ee2.(4'" m r rr
11'1
2
2
~
y'iii..
x=1
§ 8. Distribution of of digits digit,s in in complete complete period p,erio,d of periodic fractions 8. Distribution Let prime to to m. m. In Let;; be an irreducible irreducible fraction and qq ~ 2 an arbitrary arbitrary integer int.eger prime In writing the q-adic following infinite infinite pure pure recurring q-adic expansion of the number number ii, the following "decimal" to the base "decimal" base qq arises: arises:
~ == [~] ••, ['J ++ 0."Y1"Y2 · · ·."Yx7x· ...,
7x+r "YX+T = =7x "Yx
. .
(x (x
1), ~ 1),
(100) (100)
with a period rT being beingequal equalto tothe theorder, order,totowhich whichqqbelongs belongs for for modulus modulus in. m. Let N!:)(6 1 .••• 6n ) denote the number number of the times times that that the thefollowing following equation equation is is .
.
satisfied: "')'~+1
• • • 7x+n
= 61 • • • 6n
(x=0,1,...,P—1), (x = 0, 1, ... ,P - 1),
where p ~ T and 61 ••• 6,, where P 6n is an arbitrary arbitrary fixed fixed n-digited number in the the scale s·cale of q. In In other words, N!!)(61 .••• on) 6,,) is the number number of of occurrences occurrences of the given given block block 61 ••• On .. .6,, of digits digits of of length length nn among among the first P blocks of blocks . .
.
. .
"')'1 •• ·"')'n, 72...7n-f-I "')'2.·.')',.+1 , ,..., •.• , 71...7n,
"'YP.··"')'P+n-l, 7P".7P-4-n—1,
Complete expon,ent;al exponential "ums sums Complete
46
[Ch. 8 [rh. I,I, §§ 8
by successive successive digit,s digits of the expansion (100). formed by fonned (100). The question question about about the the nature of the distribution of digits digits in in the the period period of of the the The distribution of fraction is is closely closely connected connected with with properties properties of of rational rational exponential sums containfraction;; function. This connection is based based on on the the possibility possibility to to represent represent the ing exponential function. connection is .. . 6n ) in terms of the number quantity N!:)(6 1 ••• number of of solutions of the congruence congruence aqZ
== y + b (modm),
o ~ x < P,
1 ~ y ~ h,
(101)
where bb and and h depend . where depend on on aa choice choice of of the the block block of of digits digits D1 ." . D We denot,e denote the n • We number of solutions of the congruence congruence (101) (101) by ~P)(b,h). (b, h). .
LEMMA10. 10.Let Letquantities quantitiest,t,b, LEMMA b, and h be defined defined by the equalities t
-s—, 0.61 ••• Dn = -, qfl qR
b = [:,::]., b=
•
(2 + 1)m
h
—
[tin
qfl
Then . .
=
.
h).
Proof. Let xx be any any solution solution of the equation 7x+1
=
. . .
•
.
S (0 ~ x
.
(b,0) = 0, we obtain the the theorem theorem assertion: assertion: we
It is easy to ascertain as,cert,ain that that the theestimate estimate (106) (106) can can not not be h,e substantially subst,antially improved. improved. and m m == 22?r -— 1.1. The 2-adic Indeed, let T >1, 1, aa = = 1, and 2-adic expansion of ;; has Indeed, let qq = 2, r> period Tr period 1 a 1 =O.(O...O1)O..01.... -m = -r _ 2 1 =0.(0, .. 01)0 ... 01 ....
2'—l
Choose
01 •••
Dn
= 0 ... O.
Then we get N$:>(61 '" 6n ) =
T -
n and
1 it follows from Theorem Theorem 11, 11, that that Let also, that under under Ti Tl = 1 follows from Let us us note also,
1
Q and mm = 33a• under qq = 4 and where )8n I < 1. It is is so, so, for for example, example, under • In general, for for m m == PIal ••.. .• p.a, under fixed fixed primes primes Pv PII and and growing growing all, the magnitude magnitude Ti T} 18 bounded and the the following following asymptotic formula formula is is valid valid by by (106): (106):
. .
.
5,,)
=
+ 0(1).
Now, let let us establish the occurrence of aa given given block block of of Now, est,ablish the correlation correlation between between the occurrence of digits in period of the fractions fractions + O.'Yl'Y2 •••.'Yz 'Ix... : = [~] + •.• . .
=
and
,, a = [— a ] -—= -fit rni mi +0'1112"'lx'" ml a
a
I
I
I,
(,z+r ('Ix+r
= 'Ix) IZ)
(7x+r,=7x),
where the the quantity ml where ml is is determined detennined as 88 in in Lemma Lemma 11. 11.
Distribution of 0'(digits di,its in in complete complete period ptJriod o( p.riodic (ra,ctions of periodic fractions
Ch. 8] ch. I,I, § 8j
m1 under a THEOREM 12. If Ifqflo\m qRG\m— - ml
u.
certain cert,mn n0 no
~
51
1, then
T_nTl
T - Tl Nm(T)(~UI··· ~) = ----q;-+ N(Tt)(£ '"1 (Il
•••
u.
~ )
6,,. n0 and .. . 6,.. under any n ~ no and any any choice choice of ofaa block block of ofdigits digits fi1 61 •••
b, h, b1, proof. Determine integers t, b, bI , and h1 hI with the help help of the equalities Proof. 0.61 ••• 6n
[tm],
t = -, qR
[imi 1)m] _[tm], qn Lqj
bb=1—-1, = qR h = [(t+1)ml + h=1 qR + 1)nzi 1 hhi=[ = [(1 + 1)m 1 ] _ — 1 qR qn J
[(t
iqi
[tmil b1 ], b1=1——, qR
[tm J.
[(t
=[tm
Then obviously
q"(b—bi) qn(b - hI)
= t(m -
mt} - qn({:::} —
{t;l }),
m1 h1) = qR(h -— hI) =mm—- ml
n(f(t+1)mjf(i+1)mil+ftmilJtm _qR( {(t:~)m} _{(t +q~)ml} + {t;l} _{:':}). J
—
qfl
congruence m Using the congruence
m1 (mod q"), which is satisfied satisfied under under n == ml (modqR), which is
= {tm { tm} qR qn' 1
Therefore,
J
}
n0, no, we get
f(t+1)ml_f(t+1)mi (t + l)m} = {(t + l)ml }. {
qn q"
I
q"(b— qR (b - bbi) =t(m t (m— - mi), ml), 1) =
and, since sinceml mi\m and, \m and and (q,rn1) (q,ml)
~
1.
—
qfl qR
h1) = m — m1,
(107)
= 1,
bb == bb11 (modmi), (mod ml),
hh=:hh11
(modmi). (modmt).
But then, according according to to Lemma Lemma 11, 11,
= Multiply the second equality of (107) observingthat thatTr == Multiply (107) by ~. Then, observing mtt obt,ain obtaIn h—h1 rn—rn1 r—r1 h - hI m - ml T - Tl Tl= - - TT1= I = 1"1 = - - , rn1 m1q" n1l mtqR q" and, therefore,
..!!L1"1' mt
we
Comp,/e,te Complete expon,ent;al exponential sums
52
[Ch. I, §§ 8 [Ch.
applying Lemma 10, 10, we we get get the the theorem theorem assertion assertion Hence, applying ~) = T - Tl ~ ) fi,,) . . Nm,(.,.)(~(,1t··· + N("'l)(~ V n = -n-.- + ml vI··· on · .
.
q
.
m= = pO, p19,where where p is a prime j/', and m1 = 2, m = pP, Let us notice particularly the case q = ml = prime greater than 2. for instance, instance, under under pp = 3, greater 2. Suppose Suppose further /3 {j = 11 (it is so, for 5, 7) and 3,5,7) under n ~ n,o no compare of any any n-digited n-digited block block in in the under compare the numbers numbers of occurrences occurrences of period of 2-adic expansion of the fractions fractions ,Ia and ~. The former former exceeds exceedsthe the latter latter by one one and the same by same quantity (being (being equal to "'-:1). 2 So, for example, under rn = 27 we we get get ml m1 = 3, Tr = 18, Tl r1 = 2 and So, for example, under m = 3. and no n0 = digits of of length length 1, 1, 2, or 3 in the Therefore, the number number of of occurrences occurrences of any block block of digit,s the period of the fraction period 1
27 = O.(0'O'OOlOOlOllllOllOl)O'O... 0.(000010010111101101)00. · ·
exceeds by 8, 4, and 2, of the same exceeds by 2, respectively, respectively, the number of occurrences occurrences of s,arne block of digits digi ts in the period of the fraction 1
3" = = 0.(01)01 ... · Analogously under m m= = 25 we we obt,ain obtain no n0 = 2, a.nd and the of oFcurrences o~currences of Analogously under the number of any block of digits digits of length length 11 or or 2 in the period of the fraction 1 = 0.(00001010001111010111)00 0.(0'O'O'OlOlO'O'OllllOlOlll)O'O... 25 = ...
is 8 or or 4, 4, respectively, respectively, more more than the the number number of of occurrences occurrences of the same s,arne block block of of digits digit,s in the period p,eriod of of the fraction fraction 1
5 == 0.(0'Oll)O'O... 0.(0011)00 ... · These relations relations can be observed under pp = = 3,5, These observed under 7, nn = 1,2,3, 1,2,3, and pO ~ 125 12,5 in the 3,5,7, t,able given below. \ table given below. *
5,3 53
Expon,ential Exponential sums sums with with recurrent recurrent function function
Ch. I,,, § 9]
Table of values of
81 ... 8 -~ 0 P
-
--;-
l1
J...1
1
1 1 25
_I_ 1 125 125
27 27
2 22
10 10
2
9 9 4
11
5
11
11
00 00
2
55 4 2 22
14 13 13 14 77 77
50 50 25 25 25 2,5 25
3
6
11 11
2 2
7 77
11
0
0 00
11
3 2
6 7
1
2
7
1
11
3"3
'99
0
11
11 00 01 10 11 000
11
3 3
0
11
n
001 010 011
0 11
1010 100 101 110
0
0
111
11 11 00
27
81
1
5
5 5 5 5 5 3
11 11 11 11
00
2 2
11 49
2 11 1 1 1 1
11 10
1 1 1 1
3 3 3
00 00
3
2 2
3
12
11
2
13
00
3
12
0
3
6 6 5 5
11 00 0
13 12 13 13
11 0 11 1 1
0 11 0
12
I1
"77
2 22 2 33
Exponential sums with recurrent function § 9. Exp,onential function Let us us consider consider functions functions .,p(x) satisfying s,atisfying the linear linear difference difference equation with conconsta.nt coefficients stant coeflicients
= aItP(x -— 1) + n) tjJ(x) = ... ++ antP(x -— n) +...
(x > n). (x
(108)
that any function "p(x) determined by the recurIt is j,B known (see, for example, [11]) [11]) that recurrence equality equality (108) (108) can can be represented in in the form rence
=
+... +
where rr ~ n, AI, ... ,, Ar are distinct roots where roots of of the characteristic characteristic equation
An
= al,xn-l + ... + an,
(109)
and Pi . . ,, P,.(X) whose degrees degrees are are unity unity less less than than the :PI (x), ... :Pr( x) are polynomials whose the multiplicmultiplicity of the the corresponding corresponding roots roots of of the the equation equation (109). (109). In In particular, particular, if the characteristic equation has no multiple roots, then .
(110) (110)
where where C1,. C 1 , •••. ,, C,, On are are constants const,ants depending depending upon upon the choice choice of initial values values of the the function t/J(x). If coefficients of the the equation (108) and initial . coefficient,s of initial values values 1/J(l), ... ,1/J(n) .
.
. ,
[Ch. [Ch. I,I, §§ 9
Complete exponential sums
54
are integers, then, then, obviously, obviously, under any any positive p,ositive integer x the the function function sb(s) ""(x) takes takes on on integer values. = 1, and at . . , 1/J(n) be not > 1, 1, (an, m) = at least least one one of of the the initial initial values values 1/J(1), .... Let m > a multiple of in. m. In the the equation equation (108) (108) we we replace replace xx by by 3; + + n and and transit tr'Rnsit to to the thecongruence congruence to the modulus in: m:
(modm). ,p(x+n)=a11P(x+n-l)+ + n) + n—i) + ... +a,,,1/J(x) +
(111)
can be expressed in terms of t/J( x + Since (a1,, (an, m) m) — = 1, so in this this congruence congruence t/J( x) can expressed in + , —1, —2,..., we may extend the function 1/J(x) for + n) and, setting x = 0, -1, -2, ... , we may extend 1),.... . . , t/J(x +n) 1), integers x a; ~ 0. O. determined for for integers integers xx by by the congruence (111) and and initial A function function ,p(x) determined congruence (111) . . . ,, "p(n) (see (see [21]) [21])isiscalled calledaarecurrent recurrentfunction functionof ofthe then-th n-th order order t,o to the values .,p(1), ... modulus m, m, and the sum modulus 8um 2 ' t/J(~)
= Ee tramP
S(P)
= s=l
an exponential seenthat that under under nn = 11 exponential sum Bum with with aa recurrent recurrent function. function. It is easily easily seen sums with an exponential function. these sums coincide coincide with with considered considered in in §§ 7 sums Let us show that that aa sequence sequence of of least least non-negative non-negative residues residues of the function t/J(x) to rn'1 —1. modulus in m is periodic perio,dic and that thatits itsleast leastperiod perio'ddoes doesnot notexceed exceedthe thequantity quantity'm" -1. In fact, let let us us denote denote the theleast leastnon-negative non-negative residue residue of 1/J(x) to modulus m by 1:1:: tjJ(X)='IX (modrn), (modm),
o ~ 'rz ~ m -1.
Then by virtue virtue of of (111) (111) 'YZ+R
(modrn). +... + an'rs (mod == al "Y:I:+R-1 + ···+ m).
(112)
Consider blocks blocks of of nn digits digits with with respect respect to the base base m 1,;+1 ... 1z+n
(x =0, 1,...,rn'1).
(113)
Since the number of distinct distinct blocks blocks of n digits is equal to m'1, m R , then among among the the blocks blocks (113) there exist (113) exist two two identical identical blocks blocks
(x2 > x1).
We determine determineTr by by the the equality equality Tr = X2 We
— - x1 Xl
will show show that that under any xx ~ and will
(114) Xl
(115)
Expon,ent;al Exponential sums sums with with recurrent recurrent function function
Ch. ch. ,,I, § 9]
55 56
fact, under x ==x1 In fact, Xl this thisequality equalityisis fulfilled fulfilled by virtue of (114). (114). Apply the induction. x1. In the Let us us suppose suppose that the Let the equality equality (115) (115) holds holds for a certain xx ~ Xl. the congruence congruence (112) we replace replace x by Z + + rT ++1.1. Then (112) we Thenusing using the the induction induction hypothesis hypothesis we we obtain obt,ain 7x+r+n+1 /$+r+n+l
.. . + anl'x+r+l == all'x+r+n + ··· (modm), =all'x+n+...+an7x+l Yx+n+1 (mod = all'~+n +··· + llnl'x+l == 1'2:+n+1 m),
therefore, 12:+r+n+l 7z+r+n+1 = = 1:c+n+l. 7x+n+1. But But then and, therefore, 7x+2
7x+n+1 = 7r+r+2
. .
. 7z+r+n+1,
hence the the equality (115) hence (115) is proved proved for any xx ~ x1. Xl. By By means of of such considerations considerations weget get this this equality equality for for xx < <x1 we Xl as as well well (but in this case, 1X should be expressed from congruence (112) (112) in terms of 1'%+1, ... beforehand, and and that could the congruence could be done .. ,,1'x+n 7x+n beforehand, = 1). because of (an,m) = Hence itit follows follows that that the least residues of of the the function function "p( x) have Hence least non-negative non-negative residues have a
period T, r, where period where 11 ~ Tr ~ m n. Let Let us assume 88sume that least period perio,d is is equal equal to to m's. mn• that the least Then any any block of nn digit,s digits should should occur occur among among the the blocks blocks (113), (113),and, and, in in particular, particular, Then block of . .00 being formed by zeros zeros only onlyisispresent presentamong amongthem. them. But then the block 00.... then by by (112) (112) all terms of the sequence will equal equalzero zeroand and its its least least period equal 1, all sequence of the residues residues will that contradicts contradict,s to the the assumption. assumption. Therefore, Therefore, the least least period period of the function function t/J( x ) does does not exceed m n -— 1. sequence of the least least nonnonHenceforward Henceforwardwe welet letTr denote denote the least period of the sequence negative residues of the function negative residues modulusm. m. It It is is easily easilyseen seenthat that Tr is is the function ¢( x) to modulus period of fractional of the the function .function tP~): period fractional part,s parts of
= 7z+r =
I
7x
=
rn
Therefore, the sum
S(r) =
rT
Le
2. "'(x) 'll"tm-
x=1
is a complete exponential sum" sum. Since is Si.nce under integer a
fin'
fa(x+rfl r
1
—
faxl
then by (28) under any integer a the sum ax
D
e
is aa complete sum as is as well. well.
complete exponential Comple,te expon'ent;a' sums sums
56
[Ch. I, § 9 [Ch.
recurrent functions functions satisfying satisfying the the equation equation (108) (108) and de(si),. ,tPn(x) , Let tPl(X)"" be recurrent termined by initial initial values values . .
x=j, = i,
if
3:
if if
1 ~ a: ~ n,
x
~
(j
j
= 1, 2, ... , n).
It is i.s easy to show show that
+ 1)""1 (x) +···+ + ... + t/J( z + + z) = t/J( z + + n )tPn(x). t/J( x +
(116)
In fact by by virtue virtue of of the the linearity linearity of of the the equation equation (108), (108), any any linear linear combination combination of its solutions is aa solution too. In it,s solutions i,B solution too. In particular, part.icular, the the sum sum in in the the right-hand right-hand side side of of the the equality (116) is a solution of the equation (108). (108). From the definition of the functions , n the initial values that under xx = 1, 2,. .. ,n of this this sum are equal to tPj(:J:) it is seen tha.t 2, ... values of + 2),. . . ,, t/J(z + + ii), + z) has tjJ(z + 1), 1/J(z + 2), ... n), respectively. respectively. The The solution solution 1/J(x + has the the same same + 1), initial values. But But solutions, solutions, which which have have the same same initial values, values, coincide. coincide. Hence the equality (116) is is proved. proved.
=
THEOREM 13. Let 1/J(x) be be aa recurrent recurrent function function afthe of the n-th n-th order to the modulus in, THEOREM 13. m, r. Then be its its least least period, and P ~ T. Then we we have the estimates estima,t,es r be
T
r 2 . "'(~) ~
L-J e
>
11"
-;;;-
~ 2 · ,,(:t)
!!.
n n
L..Je 71"-;;;- ~m2(1+nlogm). m2 (1 + n log m).
~ m2, m2,
z=l
:.r:=1
Proof. Since Since under under an integer integer a the the sum sum rr
Le
Sa(r) =
211'i
ax (""(3:) +.!!) r
m
3:=1
is a complete sum, then under i.s under any any integer integer zz
r
.fv,b(x+z)
(,p(x+z)
r
m
Squaring and summing suuuning over over zz yields yields
r L Le (",e +
r—1 r-l
rlSa(rW =
r
x z) 2 '. (e,&(x+z) 11'1
z=O ~=o
x=1 %=1
m
-;;;-
ax\ 2 +82:)
2
T
.
(117)
57 67
Expon,ential Exponential sums sums with with r«urrent recurrent fu-n,ction function
Ch. ch. I, § 9]
least non-negative residue of of the the function function We let 1': 'Yz denote denote the least non-negative residue We
t/J(z) to modulus modulus
m. in. Then by (116) + 1)""1 (x) + + z) = = tjJ(z + +... +n)v5,,(x) ,p(x ... ++ tjJ(z + n)t/J,,(x) v5(x + (modm), == 1~+11/Jl (x) + ... + 1z+n 1P,,(x) (mod m), and, therefore,
rr y,'(s) n ~ 271'1L..Je m = ISa(T)I IS.(r)1 ~ m 2 • = I
(118)
.2:=1
1,.... . . ,, T,- — modulusm, m, then then under under zz == 0, 1, Since r'r is the the least least period period of of 1% to modulus - 1 1 all n digits are distinct. Therefore, blocks 'Yz+l '1z+1 ... Therefore, extending the summation to to . ;.r+n of n . of nn digits, we obtain . . . z,2 all possible blocks %1 ••• all Zn of .
I
r
p—I
ax\
2
m
TISa(T)I2 x=1
rr
=
~
L..J e
m-l
a(x—y) .4(X-II) 2,ri 211'1--
L
r
x ,y=I z.,=1
"("'1(.1:)-.1/11(')
e
2 11"1
m m
%1
+ + t/J,,(Z)-.pR(J/) '""
m m
ZR
)
%1 ' •• "'Z" =0
r
~ rn n
L
=m9', bm [1/Jl(X) -1Pl(Y)] · I. b,n [tPn(X) - tPn(Y)] = mnT,
(119) (119)
Z I z,,=l
where
is the number T number of of solutions solutions of of the the system system of of congruences congruences T is
~.(~~~.~(~~} 1
(modm),
1 ~ x,y
~
r.
(120)
tPn(x) == tPn (y ) Let us us assume assume that this this system system has has aa solution solution with y Y ~ x. Wiihout Withoutloss los'S of ofgenerality, generality, we may may assume assume that that yY >> x. we x. Using Using the theequality equality(116), (116), we we get get
+ n)"pn(x), = t/J(z -— x + + 1).,pl(X) + ... + + tP(z -— x + tjJ(z) = t/J(z + y - x) = t/J(z - x + l)1/Jl(Y) + + tjJ(z - x + n},pn(Y). Hence by by (120) (120) it follows that under Hence follows that under.any .any integer integer zz
1/J(z + + y -— x) == t/J(z) (modm). is the least period, But then theny'II— - x is a period perio,d of 1%, and since since rr is period, then theny'II— - x ~ r, which leads to to a contradiction. leads contradiction. Thus, Thus, the thecongruence congruence system system (120) (120) has no no other other solutions solutions except for for solutions solutions with with yy = x and, therefore, T = T. r. Now from from (119), (119), we get except therefore, T
rJSa(r)12
m"T = in"r,
n
ISa(r)1 ~ m 2 , ISa(r)I
(121)
Comp,/e,ta sums Complete exponential sums
58
[Ch. I, §§ 99 (Ch.
= 00the thefirst firstassertion assertionofofthe thetheorem theoremfollows: follows: Hence under aa = rr n . t/J(:t) 11 LJe2"& -m = = ISO(T)I lSo(r)I ~ m 2 •
~
z=1
followsimmediately immediatelyfrom fromTheorem Theorem22 and and the The second assertion of the theorem follows estimate (121): r
az\
(1 +logr)
max
x1
x=1
n
= max IS.(T)I(l = 1~a~r
+ logr) < m 2 (1 + nlogm).
Note that that in in the thegeneral generalcase case the the order order of of the the estimation estimation r . f/J{z) 211'1L..-,e m
~
n -
~m2
z=:l
from the theory of finite be improved improved further. Indeed, Indeed, using using considerations considerations from can not be fields (see, (see, for forinst,ance, instance, [33J), [33]),ititcan canbe beshown shownthat thatunder underany anyprime primepp> ~fields > 2 and
positive integer integer n
e
Tl
•. ,,(21:) 2irs 2 71'1--
" P
(
22
Tt
+ +
z=1
= =
~
LJe
• 2irt 271'&
1/1(2a:+l)
,"
2) 2
z=l
It-I,
Tl Ti
%1 .... "n=O
:1:=1
>:I Ee E z=I
•.
21r1
1 (125)
.ll1J1t(2s)+... +.a'n tPn(2:1:) 22 l'
,
________
Complete exponential sums Complete sum's
60
[Ch. [Ch. I, 5§ 9
,
indicates the deletion where the sign sign ', in the the sum sum "".. deletion of n-tuple z1,. Zl, ••• .. ,Zn, LJ.,l.·· ....." zeros entirely, entirely, from from the the range of of summation. summation. Let T1 formed by zeros T 1 denote the number number of of solutions of the system system of of congruences congruences 40'
~.(~~).:.~l.?~.)}
(modp)
1 ~ X,y
= tPn(2y)
tPn(2x)
In the same a,arne way way as as in inthe thesystem system(120), (120),we wehave haveT1 T1
= T1
rj
p—i
~ T.
and, therefore,
2
=p'2T1
>
x=i
But then (125) (125) can be rewritten in the the form form
/
Tt
Le
rj
•
t/J(2~) 22
2'J1"1--
P
2
2
.
Le Tl
+ +
• 271'1
"'(22:+1) 22
"
P
>
x=l
pn+l == = Pp"n -— Tl = - .. 2
x=l
(126)
We We determine 18*(Tl)' equality (r1 ) with the help of the equality
IS*(Tdl = max
/
— " ~e >e rj
Tt
"'"
(
.
t/i'(2z) "'{2:t)
2'J1"1 - -
p
Ti Tl ,I
•
2ira
~e 2'J1"1
"'"
t/t(2~+1) ) p
•
Then from (126) (12:6) we we get get • 12 IS (Tl) ~
pR + 1
-4-'
IS*(rl)I >
Hence by > 22and > 11 there there exists a by (117) (117) itit follows followsthat thatunder underany anyprime primep p> andany anynn> function of of the the n-th order to the modulus such that that for recurrent function modulus pp such for the exponential S*(r) sum S*( Tl)the thefollowing following estimates estimates
1
!!.
"2 p2 < IS*(Tdl hold.
!!.
~ p2
________
61 61
Sums of of Legendre's S.ums Legendre's symbols
Ch. I, § 10] ch. 4
io. Sums § 10. Sums of of Legendre's L·egendre's symbols symb,ols Let Pp>> 22 be = an a0 + alx polynomialwith withintegral integral Let be aa prime, prime, 1(x) f(x) = a1x ++ ....., + + anx R be aa p,olynomial coefficients,n n (a1,p) > n be be a prime, prime, (aI, p) = 1, Consider one of such special cases. and
(xi' +aix x=1
show that Let us show
~ ( zn :
alZ )
I~
(133)
(n - I)JP.
In fact, complet,c fact, since under z ~ 0 (mod (modp)p) the the linear linear function function zx zx runs runs through aa complete residue system system modulo modulo pp when when xx runs runs through a complete residue residue system system modulo modulo p, p, residue then P /flIZi-rcZiZZ ç—fZX
x=1
Therefore,
Iun(alz n- l )/= I
=
~(znzn:alznz) ==
t,(zn: alz ) ==lun(at)l. Iun(ai)I.
and summing summing over z, we we obtain Squaring this equality and
1)Ian(ai)12 (p (p-l)lu n(al)1 2 =
,-1 P—i
p-l p—i
~=1
~=1
Elun(alz n- 1 )1 2 == Et('x)lu n ('x)1 2 ,
where teA) is the number number of of solutions solutions of of the the congruence congruence n 1 alz -
== A (modp).
that Since t(A) teA) ~ n -— 1, then from from (134) (134) it follows follows that 1) (p — -1)
lun (al)1)122
1) ~ (n — -1)
p—i ,-1
E IU (A)1
2
n
~=1
=(n-l)?; ztl (zn;,Xz)
(yn;,Xy)
x,y—i
= -1) 1) = (n —
E (:l: t Y)
~=1
p
x,y=1 z,y=1
+ (.\ + (A + x"-l) (A + yn-I). p p P
P
Hence, using the equality equality (132), (132), we we get the estimate estimate (133): (133): Imn@i)12 lun(at)l 2
— yfl_I) ~ n=~ ,-1 E ( zy ) [p8p (zn-l_ -1]1] y n-l) —
I
p
£ (?)
:1:,11=1
(n—l)p = (n = __ ?p P Icrn(ai)I lu,.(al)1
~ (n
P
.E (X )6 (x z.,=1
-l)JP.
Y
p
p
n-
1_— yn-I)
~ (n _1)2 p, —
(134)
[Ch. [Ch. I,I, § 10 io
Complete expon·ential exponential sums CompJe·te
64
Tinder odd n Under odd
~
for the general case also: p) = 11 the same estimate holds for 3 and (an, (an,p) also:
t
(ao + alX + ... + anx
~=l
~ (n -1)y'P.
n )
(135)
P
For n = 33 this under an arbitrary nn ititfollows thi.sestimate estimatewas wasobtained obt,ainedby by Hasse Hasse [13], [13], under follows from more general results of A. Weil [48]. One can acquaint with elementary methods result,s Weil [48]. One can acquaint with element.ary for obtaining thesums sums(135) (135) by bypapers papers[35], [35], [42], [42], and and [31]. [31]. for obt.aining estimates estimates of ofthe for polynomials polynomials of of the the second As it was was shown shown above, above, sums sums of of Legendre's Legendre's symbols symbols for degree Gaussian sums. sums. Let Let us us show show that that Gaussian degree can be evaluated with the help of Gaussian used in in estimating estimating the thesimplest simplest incomplete incomplete sums sums of of Legendre's symbols: symbols: sums can be used p
o(P) u{P) = = ~ (;)
(P < p). The availability availability of a nontrivial estimate P E . (:.)
z=1
p
e2lrif(x) 2
22
I(z)
x=1 ,;=1
= = 1.1.
P1 Pt
L >
= =
e21ri [/(,)-/(x)]
x,y=1
= = PI P1 + +L
e21ri [/(1I)-/(x)] + +
Le
21ri [/(,)-/(x)]
x>, x>y
x
~P P1 t
21ril /(z+II)-/(:C)] •
y=I 11=1
x=1 z=1
Hence, after after interchanging interchanging the the order of summation, summation, it follows that Hence, follows that 2
i'1—i
x
1
z
"2
2>
e
Ui
1
P11
P1—UI
Pi +2 > I
x
0
Ui
I
Raise ine-quality to to the thepower power2k—1• 2"-1. Then Raise this this inequality Thenaccording accordingto to (142) (142) we we obtain obtain
L
2 k—i
22"
PI P1
e21ri /(x)
Ee2T;f(x)
>2 >2e
(146)
.
yi=0 x=1
:1:=1
Applying the the inequality (146) to its right-hand that right-hand side side in in succession succession and observing that Applying Pi = = P and P" ~ P, we PI we arrive at the the assertion assertion of of the the lemma: lemma: 2
p1
Z
1
k
P1—i
k
(
Pk+1
>2 >2 e
>2 P
1
51
P1—I
0
Uk
0
Pk—l
> •••
si=0
f(z)
>2 >2 e
Yk=°
x
1
vi
1(z)
z=1
, x,,, . LEMMA 13. Let Let A A and x1,. Xl, ••• X n be b,e positive int,egers. by Tn(A) the number integers. Denote by of solutions of the equation equa,tion Xl .••• X n = A. A. Then Then under any we have any c6 (0 (0 ~P.
equations § 12. Systems Systems of e~quations for the estimation of complete rational rational sums consists A method proposed proposed by by Mordell for sum to to the estimation of the mean in the reduction reduction of of the estimation of an individual sum value p
p
2k
p
s=1
a1=1
under kIc == nn or, in other other words, words, to to the the estimation estimation of of the the number nmnber of of solutions solutions of of the the system of of congruences congruences
~l.~:::.~.~~.~~.:.:::~~k. }
(modp).
xi +··· + xk == yf +···+ Yk
Similarly, in Vinogradov~s Vinogradov's method method the the estimation estimation of of Weyt's Weyl's sum sum is is reduced reduced to the Similarly, in the estimation, under aa certain cert,ain Ic, Ie, the mean value of the quantity 2k
P
L:P e
+
27ri (al x + ... a ,.zR)
s=1
being equal equal to being 1
1
J. .J f···J o
.
0
P
'L.i " e21ri (O'lZ+",+ct"x") Ee2w1 x=1
2k
dal·. .. dan .
Systems Systems of of equations
Ch. II, 1/, § 12] 12]
79
coinciding with with the numb'er number of solutions of the system and, as it will will be shown below, coinciding system
of equations equations
XI+...+XkY1+...+yk)
~l.~:::.~.~~.~.~~.:.:::~.~k.}, xf + ···+ xi:
=
1
xj,yj
(157)
P.
yf + ·· · + Yk
flenote by by S( S(ai,. . ,, a,,) Denot,e at, ... an) the Weyl Weyl sum .
p 21fi X a,n ) = -- "" .,a,,) S( "" ~l,oo 0' L.J e (Ol:.r:+..• +an ") ·
2:=1
fixed integers, integers, and N~~)(Al'.'" , k,,) Let n ~ 2, '\1,. . . , An be fixed An) be be the number number of Let integral solutions of the system of integral of equations equations ' 0
•
,
X1+...+Xk—(Th+...+Yk)A1
) .~l.~:::.~.~~.~.~Y,'~'~"""'~~,~~.~.~.l,' },
(158)
xf + ... + xi: - (uf + 0" + Yk) = An which the the quantities quantities Xj x3 and and 1Ij y, vary within the limits in which 1 ~ Xj ~ P,
1 ~ Yj
~
(j=1,2,...,k). (j = 1,2, ... ,k).
P
with the under A1 Obviously, under Al = ... . o. == A,, An = 0 this this system system of of equations equations coincides coincides with system (157). Let us consider consider the simplest properties of such 5yst,ems. systems. First of Let of all all we we shall shall show show that under under any any positive positive integer integer kk we we have have ru ru )1 2k -= ,a,,)12k 18( 15(ai,.. ~1 ~n ,.
0
.
•
""
L...J
,
... "/P,>(\ 1V k,n AI,·
. 0
•
,
\,
An
)e2ri(01~1+... +an~n) ,
(159)
~II."I~"
where the the range of summation is where lAvl < kP" I~"I
(ii (v
1,2,... = 1,2, ... ,n).
Indeed, since Indeed, p
Sk(ai,. ,a,,) = . .
L X1,...,Skl
e21ri (al (XI +... +%.)+... +a,,(x~+ ..• +z;»
,
then, obviously, obviously, 2k = IS( at, ••• ,(tn)1 ,an)12k = S(ai,. . .
P
L Zt.···,lIk=!
e21ri (01(2:1 +'.. -'k)+... +OR(X~+ •••
-,:».
(160)
[Ch. /1, §§ J2 (Ch.
Weyl's Weyl's s,ums sums
80
Here we unite addends with fixed values of the sums xi
1,2,. .. ,n)· +... + ... —- Yk (v (II = = 1,2, I
••
51+...—yk=A1
~~.
x~
:. : : :
~ .~k. ~.~~
}.
+ ... -Yk = An
P (j = Since 11 ~ Xj ~ P and and 11 ~ Yj ~ P = 1,2,.. 1,2, "".,k), k), then by (158) the the number number of ofsuch such .. ,, >.,,) and besides addends equals N~~(>'lt •••
1,2,...,n). (II = 1,2,. ,n). (z,=
IA" I = Ixr +···+ xi - yf —...—yfl< - ··· - Yk I P
= -
e2 11'i (at (Zl + ... -,.)+... +Q,,(z~+ ... -,;» '""
. )(\ L..J N(P. k,n 1\1,· -"1,...•-""
\
. I
•
,
I'\n
)e2 11'i(Ot-"t+...+ a n-"n)
I
The equality (159) (159) follows follows by (160). , 2k in the multiple The relation (159) is is the expansion expansion of ofthe thefunction functionIS( IS(ai,. at, ... ,(tn)1 multiple . Fourier series. series. The quantities N~~(>'l" .•. ,, >'R) are its Fourier coefficients. coefficients. Therefore, .
~~(>'h •· •, >'R) .
-f f IS(~ 1
-
1
=o
~
)12ke-211'i(al-"1+ ... .. '-Al"."u",
.,.
+Qn'\")d~'-Ao!"''-An .. d~
(161)
0
and, in particular, 1
1
f . ·f
,o) = ~~(O, ... ,0) J.. .
=o
.. , a R )I 2kda t .. .. • daR' IS( £lh' ..
0
Hereafter we we shall shall often often use use the the a.bbreviated abbreviated notation Nk.n(P) and Nk(P) instead of (P) ( ) .
( •instead . • of N Ie(P) the modulus Nk,n 0, ... ,0 , and N k AI, ... , An} Instead ,,.. (At, ... , An)' Since SInce of the the modulus of the the integrand, integrand, then itit of an integral does not exceed exceed the integral integral of modulus of follows from from (161) (161) that that under follows under any any A1,. At, ..... ,, An ,An) ~ N~P)(>'t, Nr(A1,..... '>'R) .
1
1
f ... f IS(at, ... ,a,,)1 2kdat .. . daR == Nk(P). Nk(P). .
o
0
. .
(162)
ell. II,1/, § 12] ch.
Syste,ms of of equations equations Systems
81 81
Let us show the the validity validity of ofthe thefollowing following equalities: equalities: Let . LN~P)(Al'" "An) = N~P)(All",IAn-t}, >
(163)
.
~n
L
~P)(At, .. " A.) = p2k ,
(164)
~l'''''~R
A)] = N2k(P).
(165)
In fact, according to the introduced introduced notation not,ation the thenumber numberof ofsolutions solutions of of the the system system of equations
X1+...—yk=A1
••••• n-l
xl
~~
.-:.......
+• • • -
~ ~~.~ .~1
1
(166)
••• }
n-I
Yk
=
\
I\n-l J
is Complete this system by the equation An_i). Complete equal to N~P)(Al"'". , An-I). is equal .
The number of solutions solutions of of the the completed completed system system equals equals N~P)(Al"'" , An). Every solution solution of the system (166) (166) satisfies satisfies one and only one of of completed complet,ed systems syst,ems arising arising values An, and every every solution solution of the completed completed system satisfies satisfies the under distinct values system (16 (166). . ,, An) extended to all system 6). Therefore the sum sum of of the quantities quantities N~ P) (A 1, ••• is equal equal to the number of the the system system (166), (166), Le.) i.e., the possible values An is number of solutions solutions of .
.
1
1
.
equality equality (163) is valid. The equality (164) (164) follows follows immediately from from (l'6,3):
L
L
. A.)= = Nt)(At"",
~l, ••• ,~n
. ,A,,_1) = ... N~P)(All,,,,AII-d= .
~l .... ,~n-l
= p2k = = L~P)('\d = p2k. A, ~t
prove the equality (165), To prove (165), we we consider the system system of equations
Xi+...+X2ky1...y2kO)
~~.:.:::~~~~~~1.~:::.~.~~.~.~}.
xf +··· + x~k - yf - ··· •
y~k = 0
(167)
J.
The number of solutions of this system is N2k(P). The N 2k (P). Collect Collect those those solutions, solutions, for for which which , under fixed A1,. At, ... the equations . . ,An
Xi+...ykAi
~~ .~ : : : ~ .y..~ ~.~~. },
xf + ... -Yk = An
Weyl's sums Weyl', "urns
82
[Ch. II, 11, §§ 12 12 [Ch.
~~~,1, ~ ,~,~~ ,~ ,~~~~,~, ~ ~,2,k, ~ ,~~ } :::
n
" ""
.fl
n
-r Y2k —An n Y:+l -r + ...···+ Y2k— - X:+ 1 - · · .· — - X;k = i
i
(P)
2 fulfilled. Obviously the (A1,. . , An)] are fulfilled. the number number of of such solutions is equal to [Nk [Nt>(..\l"'" ..\n)t To each n-tuple . . ,,An there corresponds To n-tuple of ofvalues values A1,. At, ... corresponds one one definite definite aggregate aggregateof ofsosolutions of the system system (167) (167) and and each each solution solution of the system syst,em enters into into one one and and only only A,, we get nfl Thus,considering considering all all possible possible n-tuples n-tuples A1,.. At, .... ,, A,u all one of these aggregates. Thus, (167) and, and, therefore, therefore, solutions of the system (167) .
.
= N2k(P).
In investigating properties of of the the system system of of equations equations
~~,:,:::~ ~k ~,~~
)
.•
x~
+ ... -
y~
(168)
} ,
= An
and in deducing estimates of of Weyl's Weyl's sums, the relationship rela,tionship between between the the exponential exponential sums p
a ) - " e 2ri (OI:1:+ ••• +O'n:l:") 5(""t"",n-LJ ' ;'\1
2:=1
considered as as functions functions of nn variables a1,.... . . ,, an and the number of solutions of the considered variables at, equation system essentially. This relationship is seen from from the expansion equation system (168) (168) is used essentially. 2k , a multiple Fourier series: of the function IS(ai,.. of the function IS(al," ., a n )1 in Fourier series: .
2k = IS( ,lt n )1 IS(ai,... = lt ll ...,an)12k
00
L
. Nt>(..\1,"',..\n)~2 ... i(Q'1~1+ ... +OIn~n>,
(169)
'\1 •...• '\n=-OO
Actually the the series (169), as as it was in the equality Actually series (169), was shown shown in equality (159), (159), is aa finite finite sum, sum, because ifif at at least one of quantities A,, in absolute absolute value valueisisgreat,er greater than than or equal to because All in kP", then kpll, thenthe thesystem system(168) (168)has hasno nosolutions solutions and andthe thecorresponding correspondingFourier Fouriercoefficient coefficient vanishes: =0 N~P)(Al, ... ,An)=O .
.
(IA"I~kP"). (IA,,I
We We shall show show that that the theabove aboveestablished est,ablishedproperties properties(163)—(165) (16,3)-(16,5) of of quantities quantities (P) . . . are evident corollaries of this expansion. N~P)(Al" corollaries of . . ,, An) are Nk (A1,. .. In fact, fact, setting settingQla1==... In ... = an = 0 in (169), (169), we obtain the equality equality (164): (164):
p2k p2k
=
L ~1'.",'\n
N~P>(..\ .. ,. · , ..\n)'
equations Systems of of equations
Ch. II, §§ 12]
83
equality (16,5) (165) follows follows atat once identity for for the The once from from Parseval's Parseval's identity the function function The equality IS(al"" ,an )1
'E >2
2k
:
[Np(A1,.. A)]2 = [N~P)(>\}, .. ·,.x,,)f =
~1'···'~"
1
1
j...
j[IS(al, [isai,.... ,an)12kfdal ... dan J. J 0 0 .
o
.
o
1
= =
.
. .
1
j ... j
o ....
, IS(al," ...,an)14kdal" IS(ai,.
.dan == N 2 k(P).
. . .
0
Finally, (169), we we get p'inaily, setting an = 0 in (169),
S( ~ IIS(ai,.
~
""'1 , • •. .• ,, "",,.-1
)1 2k =--
"" L-J >2
Ai,...,A,—t ~".",~n-l
[""N(P)(\ \ )]e2""i(Ql~1+ ... +a"_1~n_l) • L-J k AI, • • •. , AR ~"
)1
2k uniqueness of ofthe theexpansion expansionofofthe thefunction functionIS( IS(ai,. at, ... , Hence by virtue of the uniqueness . . ,(tn-l
in the Fourier series series
IS(ai,. .. ,an_i)12k =
..
>
(163) the equality (163)
> follows.
important question question in in the thetheory theoryof ofthe thesystems systemsof ofequations equations T'he most important The
+ -!lk = 0 } .... , xf + - y: = 0 Xl
charact,er of of the the growth growth of of the the number nwnber of of system system solutions solutions is aa question concerning the character in dependence dependence on int,erval of the variation variation of of variables, variables, i.e., aa on the magnitude of an interval question concerning concerningthe thecharacter characterofofthe the growth growthofofthe the quantity quantity Nk(P) Nk(P) while while P question increases infinitely. It is x, ~ P (j = is easy easy to to establish est,ablish aalower lower bound bound for for Nk(P). Nk(P). Indeed, Indeed, since since 11 ~ Xj = 1,2,.. . ,Xk 1,2, .... ,, k), the quantities x1,.. Xl, ••• can be chosen in pk p k ways. Choosing Choosing then then !II yj = , obtain pk Therefore we we have have the the estimate Zl, ••• = Zit Xk we obt,ain pic solutions. Therefore •..,, Yk =
Nk(P)
(170)
[ch. [Ch. I!, 11, § 12 12
Weyl's sums
84
Next, by (162) (162) and (164)
p2k = p 2k =
L
.
.
~t ""'~Q
n(n+l) n(n+1) 2 -Nk(P), 1 1 ~ (2k)n p-2 Nk(P),
L
N~P)(Al, ... ,An)~N,,(P) ( Nk(P)
~l.···t~n lAp l (xj -— yJ) = —
(y1
as-II
= ,,=0
;=1
j=1
C:
where of ss objects objects vv at a time. where C' denotes dcnot,es the the number number of of combinations combinations of time. Therefore each each solution solution of of the the system system (175) (175) isis aa solution solution of of the the system system (174). (174). It isis is aa solution just as easy to verify verify that in in its its turn turneach each solution solution of the system (174) (174) is solution of the the system system (175). (175). But then then these these systems systems of of equations have the same same number nmnber of of solutions, and and this is what we solutions, we had had to to prove. prove. Note. No,t,e. According to Lemma Lemma 15 15 11
J 1 1
Nk(P) = f··· L Nk(P)=J...J o
0
2k
a+P n t?lI'i(a z+ ...+a n :t ) ,
dXl .. , dX n1
x=a-F1 %=4+1
and, therefore, under under any anyinteger integer aa the the equality equality
J J '" 1 1
1
• ••
o
L...,
0
=
2k 2k
4+P e21ri(OlZ+...+0"s")
d~wI
..J_
• • • UZn
2:=4+1
j... j t
o
0
$=1
e2 11'i (a,z+...+anz
n
) 2k dXl
••• dX d n
[Ch. [Ch. II. II, §§ 12 12
Weyl's sums sums
86
holds. 7), Tk(P) Tk(P) be the Let, Lemma 7), the number number of of solutions solutions of the Let, as in §§ 6 (the note of Lemma system of s'ystem of congruences congruences
.~~.: ~.~~.~~...~~:!. }, X1
z~
+ ...
+
— ilk
(modp)
0
(176) (176)
- 11: == 0 (modp")) (modpR)
solutions of of this this system can be expressed in We shall show that in" terms terms We shall that the thenumber number of solutions (P) of the quantity N~P) (AI, · · · , A.). LEMMA16. 16. We Wehave have the the equality LEMMA TI:(P)= Tk(P)=
2:
>
N1P>(Alp, ... ,A R p R ),
~l, ••• ,~n
where summa,tion is extended ext,ended over over the region where the summation Ad
> n", p be LEMMA17. 17.Let Let n ~ 2, LEMMA be aa prime, prime, pfi ~ P p < 2Pn, and pfl_l. p1 = P1 p"-I. Then Then under k >>n2 n 2 for for the the number number of ofsolutions solutions of the system syst,em (178) (178) the
=
=
estima,t,e estimate
,,(n+1)
Nk(P)
2
h,olds. holds.
Proof. Let Let f(x) f(z) = QIX +... QnX"and andthe the sums sumsSS and and S(z) 5(z) be Proof. be determined determined with with + ... + + £tnx" the the help help of the equalities p+pPi p+p Pl
L
S(z) = = 5(z)
SS = = > e21ri /(Z), r=p+1 -==1'+1
P1 PI
Le
21ri /(Z+pi).
2:=1
Then, obviously, P p
s= L 5=
Pj Pt
Le
P P
21ri
/(.r+,,:i:)
z=1 &=1 *=1 z=1
= = LS(z), z=1 %=1
2k = 1512r1512k—2r 1 ISI 2r ISI2k p2k_2r_IJSf2r 181 = IS1 2 r1S12k-2r ~ p2k-2r-
P
L E IS(z )1
2 1:-2r.
*=1
Since the number of solutions of the s'ystem system (178) (178) grows growsas as PP grows, grows, then then using using the equality (161) (161) and and the note obtain equality note of of Lemma Lemma 15, we we obt,ain 1
NI:{P) :s;; NI:{pPtl
1
= f··· f 181 21: dOll • • • dan o
0
[ch. II, § 13 [Ch. 13
Weyl's sums sums Weyl's P
~ p21:-2r-l L
f··· JISI2rIS{z)121:-2rda 1
1
[• I
r=1 0
ISI2nIS(z)12k_2rdai1
'".
.
(179) (179)
dan. daR'
0
Let the maximal value of summands summandsininthe the sum sum (179) (179)be beatt,ained attained at at z = value of = zoo Then we obt,ain obtain we
Nk{P) Nk(P)
~ p2k-2r
f ...•JJ L 1
>
o
2r 2r
P+pPl p+pPi
1
chi f(z)
x=p+1 ~=,+l
0
2k—2r 2k-2r
p1 P1
?=
e hi f(%o+,:i:)
dal da1 ... , dan. .
.
z::::l
seen that that the It is easily seen the integral integral in in this this estimate estimate is is equal to the the number number of of solutions of the system of equations 1 .~,~.~:::~,.~r. ~ .(.z.o, ~ ~l.),. ~::: .~.~~. ~ ~lI,','~~:~', },
= (zo + p'Xl)R +,. -- (zo + PYk_r)n P < X j, Yj ~ P + PP1 , 1 ~ xj, iJj ~ PI,
xf +., --
y~
or, this is just just the the same, s.ame, to to the thenumber numberof ofsolutions solutions of the system
or,
+ x1) + ... .. ~~~ .~.~~ ~ ~ . . . . ~, ~~~ .~.~~~ ~,{.~. ~ :.~l.~.~::: .~.~~o. ~ ~~k,'~~ ~ (zo
= (zo
— (zO + yr)
(ZO
+ Xl)R +, .. -
+ pthj) +... — (ZO +
P?'k—r)
.. } ,
+ Yr)R = (zo + p Xl)R +... - (zo + PYk_r)R p—zo pZo <x,,y1 < Xj,Yj ~p- Zo + pP1 , 1 ~ Xj,Yj ~ PI, (zo
1
In its turn In turnthis thissystem systemisisequivalent equivalent (see (see Lemma Lemma 15) 15) to the the system system
.~~ ~"":.~.~,~.~.:.,~~~,~. ::.~.~~~,~!, }, R+ • • , - YrR= PR('n+ Xl ' •• -
3:1
p—zo P - %0 <x,,y1 < X j , Yj ~ p Let Let
Zo
+ pPl ,
'n) Yk-r
11 ~
xj , iJj
~ PI-
us replace the y, by a wider the interval interval of of variation of x1 Xj and 1/j wider one: (j=1,2,...,r). (j = 1, 2, ' . , , r ),
Then, collecting collecting solutions solutionswith withfixed fixed values values of of the the sums sums xi +, .'-Yk-r (ii (v = 1,2,... 1,2, ... ,,n) n) and using the estimate estimate (162), (162), we we get
Nk(P) Nk{P) ~p2k-2r
L
~:~{~"
... ,~n)N$2PP1){~lP""'~Rpn) .
.
.
.
~l,""~"
/' ~
2r '"'" p2k_2rNk_r(pl) P2k- N k-r (P) 1 L-, ~ll.",~n
N(2p P l) ( \ \ n) r AlP, • • • , AnP , .
Vinogradov's mean mean value value theorem theorem Vinogra,dov's
13] Ch. Cli. II, §§ 13]
8'9 89
r(2P1)" where the summation is extended extended over over the the region region 1'\,,1 < r(2P (II = 1,2, 1,2,... .. . ,n). 1)1I (ii Hence, since since PI P1 = = p"-l, using Lemma Lemma 16 16 and and the the note note (90), get the lemma Hence, (90), we we get lemma assertion: Nk(P) pr;-
>n
..
·
)r n1 n-l
1 (1+_
and by the induction induction hypothesis hypothesis 2k—2r—
Nk_r(Pl)
24n(k_r)Tp,
n(n+1) 2
Lemma 17 1T But according to Lemma
Nk(P)
k
2
Nk_r(PI)
(181) (181)
Weyl's sums Weyl's
90
[Ch. II, [Ch. II, § 13 13
and, therefore, by (181) (181) 24nkr+2nk(pp1)2k_
Nk(P)
n(n+1) 2
pP11 = = pH >2nlog(n to zero. zero. So under 2n log (n++1) 1)we we get get
_n(n+1)(1
1\T
1
nI 3n log log (n (n + +1) Hence itit follows follows that that for Respectively, 1) we we have ee,. r < 2(n 2(n+1) 3 log(n have the the estimate any e6 > 0 under kIe >>3n3 3n log (n + +1) and n ~ we we have
le
Nk(P)
~
2 kk22
2ft ' p P
n(n-4-1) L 2k— 2 " ,n(n+l) ---+£ 22
=0
(
P
n(n+1) n(n+l»)
2k— 2k---+e 4-s
22
•
(182)
On the other hand, On hand, by by (171) (171) 1
n(n+1)
2k —
2
from comparison comparisonof ofthis thisestimate estimateand andthe theestimate estimate (182), (182),that that the order of It is seen from the estimate estimate (180) (180) is is almost almost best best possible. possible. A question about the the least least value value of of k, k, under under which which the the estimate estimate (180) (180) isis fulfilled, fulfilled, A difficult. This question question is important i,mportant in in connection connection with with the thefollowing following is much more difficult. circumstance: estimates estimates of of Weyl's Weyl's sums sumsobt,ained obtainedby by the the help help of of the mean mean value value circumst,ance: to est,ablish establish an an estimate of the theorem are, as as aa rule, rule, more more precise, precise, if one succeeds succeeds to form (180) (180) under lesser values values of of Ie, k, i.e., i.e., the the lesser lesser the the b,etter. better. form
Vino,rado,v's m,ean value theorem Vinogradov's mean
Ch, 11, §§ 13] 13] ch. II,
91 91
Let us show show that that the estimate Let us estimate n(n+1) Nk(P) = o(P2k_ 2 +e)
(183)
cannot fulfilled under < n(;+1). n(R2+1). Indeed, under k < Indeed, according according to to (171) (171) N/t(P) Nk(P) ~ pk and, cannot be fulfilled therefore,ininorder order to to satisfy satisfy the the estimate estimate (183), (183), the the estimate therefore, n(n+l») e ph pit = 0 p2k- -2-+ (
should be fulfilled, fulfilled, but that isis possible possible only only under kIe ~ n(n2+1). Thus the best result but that which might be be expected expected to to obtain obtain is is getting estimate (with respect which might getting a precise precise estimat,e respect to to n(n+1) the order) = n(fl2+1). The estimate estimate (183) (183) following following from 15 was was from Theorem 15 the order) under under Iek = 3 log (n + 1). Using the Linnik lemma (Lemma 9) instead of obtained under kIe ~ 3n 3n3 1og(n + 1). Usi,ng the Linnik lemma (Lemma 9) instead of obt,ained
Lemma 7, we we get get now now this this estimate under kIe Lemma 7, LEMMA18. 18.Let Let n ~ 2, LEMMA 2,
pP
p ~ (2n)2n, p
[pp-l] + under kIe ~ + 1. Then under
R("2+!)
~
3n 1). 3n22 log log (n (n + + 1). 1 1
be aa prime, prime, !Pi
~ p
11
and PI P1 = = < pn, and
we have we have the estimate n(n+1)
Nk(P)
Proof. As we introduce introduce the the not,ation notation f(x) 1(x) = = alX Proof. As in in Lemma Lemma 17, we a1x S= =
p+pPi P'+PPt
L: e21fi 1(1:),
(184)
Nk_n(Pl).
2
a,,x", + ... + lrnX",
P, Pt
21ri /(Z+p:i:). S(z) = = L:ee2"
z=1
x=p+1 ~==p'+l
Then we get P
=
~
22
S(zi).... .S(zk) S(ZI) ,S(Zk) .
%t ..... zl:=1
Weshall shallsay saythat that aa k-tuple k-tuple Zt, zi,.••• . . ,,Zk belongs to to the first first class, class, if itit is possible to find We Zk belongs nn distinct quantities quantities zZj in it. All All remaining the second second class. class. remaining k-tuples k-tuples are are to be of the Since 2
=
S(zl)...S(zk)+ ,%.
~1 •••• Z1,...pZfc
~l.···.~'
~ 2 L:l S(Zl)",S(ZIr)r +2/ L2 S(Zl)' "S(ZIr) Zl J···.-'Ir
%.
Zj,...,Zk %1 •••••
2,
[rh. [Ch. II, II, §§ 13 13
Weyl's sums sums
92
of the first and L:l and L:2 are over over k-tuples k-tuples of and the thesecond second classes, classes, respectively, then observing observingthat that P ~ pP1 reS'pectively, then PP1 and using using the note note of of Lemma Lemma 15, 15, we We obtain sums where the sums
NI:{P) Nk(P)
Nk(pPl) ~ NI:(PP 1) =
1
1
2 / ....JlSI2kdai / ISI 1: 001 .. •. · dan .
= o 1
~2 /
0
1
I:l S(zl) ... 8(Zk)
... /
o 1
1
+2/.../ o o
dal ... da n
,z,
%t,••• Z1,...,Zk
o0
o
2
2
I:2%. 8(ZI)'"
(185)
S(Zk) dal'" dan.
Z,,...,Zk %1 •••••
0o
Denote by NL D'enote N k and NC the integrals of the right-hand side of this inequality. inequality. HereHereafter we shall designate the number of combinations combinationsof ofkkobject,s objectsnn at at a time by by Cr, Since Since nn distinct quantities can can be be arranged arranged on kk places places in Or ways, ways, then
Nk ~ (Cr)2
J...J I:; 1
1
o o
where in the sum
0o
2
S{ZI)'" S{Zk) dal ... dan,
Z1,...,Zk ~I"",%k
occupy the the first first nn places placesand and the the variables variables Zn+l, ... ,%1e L: ~ distinct z3 Zj occupy . .
,
p]. Hence, observing that independently run over over the the interval interval [1, [1,p]. Hence, observing 2
2
Ll 8(ZI)"
.8(Zk)
%1,••• ,%'. Zj,...,Zk
I:; S{ZI)'" S{Zn)
= =
L8(z) ES(z) Z=1 %'=1
Z1,...,Z1, %1,"·'%" ~
2(k—n) 2(k-n)
p
2(k-h)-1
~P
I:; S{zt}",S{Zn) -'I ,oo •• %'"
2 P
LIS{z)1 2 (k-n), %=1
we get
z1 o
I:18(ZI)'"
2
S(Zn) IS(z)1 2 (k-n) da t ••. dan.
It is is easily easily seen seen that under under aa fixed fixed zZ the the integral integral in this estimate is equal to the the number of solutions of the system of equations
+... — (z
.. ~~~ .:.~~~.:.'.'.'.... ~ ~~:.:.p.'~~!.~. ~~.:.~~:.:.::: ~ ~~.~~~'~~~.' ': }, (Zi +pa4) +
(ZI
—
= (z
+ px~)n +... - (t" + py~)" = (z +PX1)" +... - (z +WIt_n)n
1
(186)
Vinogradov's mean theorem Vinogra-dov's m,.anvalue value theorem
13] Ch, 1/, §§ 13] ch. I!,
93
where 11 ~ xj,yj,Xj,Yj ~ PI, 1 ~ Zj,tj ~ P and under i :I Zj, ti ~ itj are fulfilled. Introduce new variables Xj and Yj
=I
jj
the conditions
Ii
Zj
+ pxj = Z + x j,
tj
+ pyj = z + Yj
= 1,2, .. "n).
(j
Then the system (186) takes on the the form form
.. ~~.:.~.~~.:.'.'.'. ~ ~~.:,~~!.~, ~~.:.~~!.:.',:', ~ ,(~,~ ~~:.~~., }, )
(z + Xl)R
+... -
(z
+ Yn)R = (z + pXI)R +... -
(z
(187)
+ PYk_n)R
thy, determined by and the region region of of variables variables variation variation is is determined by the the conditions conditions 11 ~ x j , iJj ~ PI' P - Z < Xj, Yj ~ P - z + pPl and Xi ~ Xj, Yi ¢. Yj (modp) under i :F j. By (174) (174) the number of does not exceed By of solutions of the system (187) (187) does exceed the number solutions of of the the system of equations of solutions
+...
.~R+.~ '~"""'~'.~.~'~= ~.(~1 . . ~"""'~'~~-.'~ '>.' } , R('R+ 'n) Xl
Xl
••• -
Yn = p(thi +... — 11k—n) )
Ytin
P
Xl
••• -
Yk-n
with fixed Collecting solutions with fixed values values of of stuns sums
1 ~ Xj,Yj ~ pPl pPi + 1 +p,p, i -:; j => Xi ¢ Xj, 1Ii ~ Yj (modp), 1 ~ Xj,Yj ~ Pl.
(ii == 1,2,. xi + ... -— Yk-R (II 1,2, ... , n) +... .
.
,
we
obt,ain obtain
L >
Nt: ~ (c:)2 p2(k-.)
~~~('\l0""'\R)N=('\lP,'" ,,\npR)
~l"",~n
L
~ (Cf)2 p2(k-R) Nk-.(Pd >
N:(,\lP, ... .. ,'\.p.),
~l'''',~n
, where N:(AIP, ... . . , Anpn) is the number of solutions solutions of of the the system the number
~1. ~ '~'.~~.~. ~.1~ xr + - y: = Anp .
R
} ,
1 ~ Xi,Yi ~pPl +p, :f:. j => Xi ¢. Xj, YiIll ¢ 11j (modp),
i
and the n(Pil + 1)" the summation summationisisover overthe theregion regionIAVI 1'\,,1 < n(P 1)" (ii (v according according to Lemma 16 16
:E
T(mptz),, T(pPi1+p) N:(AIP, ... ,ARpR) = T:(pP + p) ~ T:(mpR)
~lt ••• ,~"
where
1,2,... = 1,2, ... , n).
P] + 1 ~ m = [PP1+ m = [PP1 + P] +1 pR
2PIP-(n-l) 2
,
But
Weyl's sums
94
and
[ChI II, II, §§13 13 [Ch.
(mp") isisthe T:(mp") thenumber numberofofsolutions solutionsof ofthe thesystem systemof ofcongruences congruences
x1 +... — y,,
}
.~~ .~:::.~.~.~.~~...~~~~::. , xf +... - y: == 0 (mod?)f (mo,dpR) (modp)
0
1 ~ Zj,Yj ~ mpR, i ~ j => X. ¢. Xj, Yi ¢. Yj (modp).
Therefore, using the estimate (93), obtain (93), we we obt,ain n(n+1)
2
2
2
—
n(n+1)
"
2
(Cfl2p2
NL
n(n+1)
k
1
—
(188)
2
Now we we shall shall estimate estimate the the quantity Nk'. Observing Observing that that the number Now nwnber of k-tuples of second class class does does not not exceed exceed n kp n-l, we get the second
L
2 2 , p
2
L
~ n 2k p2n-2 >2 15(z)1 2k ~ n 2k p2n-2
S(zi) . . . S(Zk) S(ZI)'"
z1
%1 , •••• %~
",=1
,
NC
~ n 2• pl fl p2fl-2 L
f·· ·f IS(z)1
%=1 0
1
p
pr L > IS(z)1 n
2k 2 - n,
.p=1
11
/
2
.-
2
dat, =—n2k fl2kp2nP2n_lNk(pj) nda 1 •. •. •. dan prp2n-l Nk-n(P1 ).
0
n(n+1) Since +1) and n 2 , then and pp>> n2, Since by by the lemma conditions k ~ n(R 2 n(n+1) 2
1
n(n-I-1)
(2k)2t*p2k
2
and, therefore, n(n-F1) n(n+l)
k " 11 ( k)2n 2n 2 NL' ~ '2 2 Hie 'Pt p —k -22- - Nk-n (P) l ·
(189)
Now we we obt,ain obtain the lemma Now lemma assertion assertion from from (185), (185), (188), (188), and and (189) (189) n(n+1) n(n+l)
k , Nil ('_)2np2n 211:- -2N Nk(P) Ie (P) ~ 2N, + 2 It ~ 2 2~ 1 P 2 HIe-n (PI) .
us to make make the the statement of the mean value The recurrent recummt inequality inequality (184) (184) enables us stronger, because because this this inequality inequality reduces reducesthe the estimation estimation of of NNk(P) theorem essentially stronger, k( P) (but not not to Nk-,,2(Pt ) 8.8 as it was obtained earlier in to the estimation for Nk_n(P1) Nk-.(Pl) (but Lemma 17).
Vinogrado,v's m,eanvalue value th,eorem Vinogradov's mean theorem
13) Chi ch. 1/, § § 13]
THEOREM
16. Let 16, Let n ~ 2, 0, kk = = 2, Tr ~ 0, eT
1 n("2+ )
95 95
+ nT, and + nr, ].\r
n(n—l)(1 = n(n; 1) (1- ~r. 2
Then num,ber of of solutions of the system syst,em (178) estima,t,e Then for for the number (178) the estimate n(n-f 1)
(2k)2k(2ny3P2k
Nk(P) holds under under any any P holds
(190)
2
~ 1.
Proof. Since, Since, obviously, obviously,
6o +... +
fl(fl_1) =
2
1
(i —
n2(n—1)1
=
Ti
1
1
2
estimate (190) (190) it it suffices suffices to then to prove the estimate to show show that that n(n+1)
Nk(P)
(191)
2
If rr = 0, 0, then then this thi.s estimate estimate takes takes on on the the form form N,,(P) Nk(P)
~
(2k)2k p2k-n
is fulfilled fulfilled by by (173) (173) under under any any PP ~ 1. 1. Apply Apply the the induction. induction. Let Let under a certain and is cert,ain n(n+1) 1 = R("2+ ) + nT ~ 0 and k Ie = the estimate estimate (191) (191) be fulfilled fulfilled under 1. Prove Prove it nr the underany anyPP ~? 1. n(n+i) +n(T+ 1). We for r +1, for T+ 1, i.e., i.e., under k Ie = = R(~+l) + n(r + 1). We shall consider the cases P ~ (2n)2"k 2 2R 2 and P 2 8(y,z)yz. >2 e 2w
'/(s+-,.r)
r=1 , •.p=1 e=l
• '=1
, •.p=1
Hence, because because fO(y, :)I ~ 1 and Hence" l8(v,z)1 Pt
n21D
—
i —
4
the assertion assertion 2° 2,0 follows: follows:
p P
P~ p2
Ee >2
f(s) 2tri [(.:)
.=1
P
Pt Pi
E >2 Ee 2 ~
x=1 .-1
p,z1 ,._=1
21fi [(1:+_,.) f(s+uuz)
+ 2aPt· +
Determine S1 51 and P1 PI with the the help help of of the the equalities equalities P-l
P
= =E >2
51 S1
E
:It
([sd] +
pi = mm
e2fti /(-+,)
y=O x=1 "=:0 .'=1
1,
---l...-
Then 5121t+1 'PI P1 ~ P and, and. therefore, therefore" p1—i
2k
p
P,—1
p
2k
si.,
>2
o.
Hence, using using the the estimate 1°, Henee" 1 we we get the the assertion assertion 3°: 3°: pP ".>2e2h1tj(5) . . . . . . . .. e 2 'J1" lea) L.-, .==1
pP
.=E
1
2k-I-i 2.+1
e2 ..i /(-)
~ ~
~-1 Pt-i L.-, >
P
"......•.•....•...
11 ,=0 y=O
.,=1 z=1
p.
__1_
e2 ... /(-+,,) + P1 _"1' q ~ 2S12k+1, 2 S··1.2'+1 , + Pi —1 L.-,
-!..-... "..............
P-l p—I
pP e2 ft'i /(-+.) >2e2lri/(x+s)
p:O
&:=: 1
E E
= 22k+1 ~ ~'+1 81 = ~t+l >
xl
2.
2k
99
Esdmates Weyl's sums Estimates of of Weyl's
if, 114] § 14) CIt. ",
given by &h'e the multiple Fourier IJEMMA functionF(Ql)'" F(a1,.. . ,a L,B,MY,A 20.20.HIfa alunc,tion Fourier expansion expansion R ) is given , 00
' .. ,a.= )= ' .L.J ~'.'.".'.' . ": .' F(ai,.. F( ··.al.· >2
C·. (". .1\, \1 •••• ,. A .\ .•" ). .•2tr'l (alA, +...+0,. A.) C(A1,... ,
.
.11 •...• 1 .=-00
satisfying the condition and sAtisfying
F(ai,. ...,0,,) ,a,) ~ 0, F{al'" then under any positive tb,en p,ositive integers int,egcrs 91, .... ,q. we have , 00
F(a1" .. ,aft) ~
E >2 .C(AI91"
91 ••• q.
"\1 .... ,A..
.. ,A"qti)e~hri(Otft Al+... +a "f.. .A.).
=-oo
Proof. Since . . ,a.) ;::: 0, then Since F(o11. F(a1, ... qj—1 91-1
, .. -1
xj=O
z,,=O
F(ai, F(alt ... ,a..) ~ L >2 .,. E F(a 1 + %1 Ct=-O .,.=~o 91
,a" + %n)
(197)
q"
q,—1
oo
= =
....
>2
>2
z,=O
>2
By Lemma 2
(.\11 1+ + ""s.) L ... L e --.t ... --,;-•• = = 91 ... 9.6'1 (~1) ... 6,. (~.). >2 ... >2
qj—11 '1-
,.-}
1
2 •
tn . •.
. . .
_.=:0
xj=D -1:=:0
.
öqfl
Using this thi.s equality, equality, we we obtain obt,ain the thelemma lemmaassertion assertionfrom from (197): (197):
F(ai,..... .,cw,,) F(Ql' ) aM) 00
~ 91 ••• q"
L >2
C{A 11 ••• , l,,)e 2ft'i(OI"\I+ ...+On 1 ")6tl (~1) ... 6,.. (~,,)
Al,••.•,An=-oo 00
= 91 ••• f. =q1
L >2
0(9'1;\1" .. , 9"A.)e21Fi (0191 1 1+•..+ 0 818A.).
A, =—oo "\""',,"\n=-oo
COROLLARY. COR.OLLAKY.
Let fez)
= 01:1: + ... + a,.:c
M
and p
5(01, ... ,aR )
= E e2rr'/(.) • • ,=1
Then under any positive int,egers Th'en integers rr
~
.,(_+1) n(n+1)
n and and kIe we we have the th,e estimate es,tim,at,e
1 P'-2-2. / LR2 IS( ."alt · • · ,a. )1 < ~ k"1P I
I'ti
rr
N(P)('O \ •••• , 0)' . 2tr"ar lA .. · L.J ." , · · · , Ar '. '., e > lA,,, I'1+ ...+On>'n) ,
>'1 f"")."
where the the range of summation is where
IA"I'1+...+ 0 R9n>',.) , n k l A l, "' ., nAn ~
>
(200)
>'1,...,>'"
where by (198) be written in the (198) and (199) (19'9) the range range of summation may may be the form form
IAIII < {fkPr klPr
ii=r, V = r, vi: r,
l~ffif if
A
or, that is is just just the the same, same, in in the the form form A1 = = ... = Ar—i Al .. 1= Ar- l ==0, 0, Observing that
IArl /cpr, lAn \. . 2I1na"l"_II1 1A.-tldP.-t •• 8n ( kP--l)'" .. ~ £1 + _._q- . . . (p + q)~ ~ 64n akP 1
t
q
1 _1+-_:
2,,' ,
):.
II · 1
P"'
~ q ~ p.-l.
_____ Weyl's sums Wey1's
102
[Ch. II, § 14
and, therefore, 2k-fl 2k+l
pP
Le
21ri
n(n+l)
~
/(X)
n3kn22k+7 p
1
-+ -2 2
Nk(P). 2n'Nk(P).
2
2n
(202)
x==l
Choose 16. Since, Since, obviously, obviously, ChooseTr = [3n log n] + + 1 in Theorem 16.
n(n -1) (12
.
.!.)r < n(n -1) = ~11 __1_, nni
2n3 2n 3
2 2n2' 2n
2n
then by (190) (190) under n(;+I) + +nr nrwe we have have under kk = n(t?1)
=
3
2k- n(n+l) +.!.. __1_
Nk(P)~(2k)2k(2n)n3p
2n 2n 2 2n
22
2 log Substituting this 4n2 log nn we Substituting this estimate estimateinto into(202) (202) and andusing using 3n2 3n2 log n ~ k ~ 4n we obtain the theorem assertion: assertion:
pP
L
2k+1 2k+l
~
e2tri I(s)
128 n 3 k2H8 z4k(2n
t
3
1i
p2H 28
2:=1
~ e3a(2k+I) p2k+I-(1-1.) , p p
Le
1-...!..
1-~
>2,2, 1(x) LEMMA 2,3. f(x) = = alx +... ... + + lrnx", /3,,(x) certain interval 22 ~ r ~ n certain rr from from the interval a
lt r
Fhrthel·let Further let P
~ q ~ pr-l
6
= -+q q2'
(a,q)=1, (a,q)=l,
= :r j(B)(X),
181~1.
and the sum swn
(pv,
((sS = 1, 2,... ,,n—i). = 1,2,... n - 1) ·
-
and under a
u
Weyl's WeyPs sums
106
extended over over those those values valuesof of yy and z (0 (0 be ext,ended sa,tisfy the the inequalities inequalities positive int,eger integer t satisfy t
1I,8.(y) — - ,8.(z)1I fls(zNl < ps
[Ch. II, §§ 14
y, zz < ~ 'Y,
P), P), which which under under aa certain certain
(s 1,2,... (8 == 1,2, ... ,n—l). ,n -1).
(210)
estimat,e Then we have the estimate n(n-l) n(n— 1)
:E 1 ~ (2n)3R P-2-+ (2n)3"P 2
1
t.
11,%
Proof. Denote D'enoteby byT1 T1 the the number number of of summands summands in the sum El. Then estimating all the summands trivially, trivially, we obtain
:E
n(n-I) n(n —1) 1
2 ~ P-2-T T1. 1•
(211)
11.%
Since TT1 is the the number of those values of yy and z, which g,atisfy satisfy the conditions conditions (210), (210), .Since t is 'then by Lemma Lemma 22, 2'2, T1 T1 does does not exceed exceed the number number of solutions solutions of the system system of of then by inequalities
1,2,... o0 ~ y,z y, z
L e I
2 '1l'i(a1(II)z+... +an + 1(II)Zn+l)
I
1/=1 2:·==1 5=115=1
I
P
= =
.,An+i)
>
Since by (163) (16.3)
..
> then observing observing that
S 1 ./ Si ~
Q n
+l(Y) depend on y, y, we we get (y) does not depend
'Pp
I
N(P)(\ \ ) "'e211'i(Ql(lI)~1+... +Qn(Y)~R) hi 1\1,· • • ,"'n+1 L..J 15=1 ~tl •••• ~"+l 11=1 I) P ""' N(P)(\ \ ) "'e21ri(Ql(')~1+ ...+Q.. (II)~n) • L..J 1:1 AI, • • • , I\n L..J
'" L..J >
I
I
= =
>2
~l".'J~"
I
11=1
I
Applying the inequality (141) (141) and using the relation n + ... + Qn(Y)A n = Po + fJl~Y +... + Pnyn, which follows followsfrom from the the definition definition of of the the quantities quantities P., /3,,and and all(y), a,,(y), we obtain which
Ql(y)'xl
A))
( >2
I
x
'"
L..J >2
~l, ••• t~n
P p
2k2 2k2
N(P) (\ \ ) ""' e2Jri (,81'+"'+,B",,") kl 1\1, • • • , An L..J y=i y=1 I
=
= p2kl(2k2—l) p2kl ( 2k 2- 1 )
L >2
'\1 t.",~n t (2k -1)V, = = p2k p2kl(2k2_l)V, 2
I
Pp
N~~) (.~11 ... I
An)
L ,=1
2k22 2k
n e2 '1l'i (P11I+ ...+Pnll ) I
(221 (2'21
ofthe th,e mean mean value value theorem theorem Repeated app,licat;on application of
Chi I!, II, §§ 15] 15] ch.
113 113
where V is determined by where V by the the equality equality (220). (2'20). Let us show show that V V ~ p 2k t. Indeed, Indeed, it is is seen seen from from the determination determination of the the Let (220)the thesum sumPIflip' +.. . + quantities p., that in in the the equality equality (2'20) IJI +... +PnIJn is a homogeneous homogeneous linear function of the quantities AI," .,, A,,: linear function An: .
by (159) (159) But then by
""' (\AI, • •. .• , An \ )e211'i ({JIPt +···+Pn,'L,,) L..J N
V -III
N{P)
,···,11"
(222) (22'2)
~1, •••• ~"
N{P) (\
""'
L..J
kl
\.
Al , • • • , An
... ==
)e271'i (PtP.l +... +,8,,1'&,,)
~1 , ••• ,~ ..
~ Nk2(P)
L:
= P2klNk,(P) N~~\\11"... I An) = p 2k t N k2 (P) ~ P2k'. p 2k t.
(223)
~l"".'\n
Now, (195), we get Now, uS'ing using the inequality (195),
15(P)12kl+1 ~ < 22k1 +1
pP
P-l p—i
L: L: e 11'i l(z+lI)
2k1 2kl
2
I
,=0 x=1
P
~ 22k1 +1
2k, 2kl
P
L: L:
e2 11'i /(2:+,)
y=O x=i ,=0 x=l
/
(p2k, + ~ ~kt+l (p2k + t
t, t,e P
P
21ri/(Z+II) 2k) = =
)
y=i x=i
~kl+1(p2kl ++ S1). 51).
Hence, using using the the inequalities inequalities (221) (221) and and (223), (223), we weobt,ain obtain the the lemma lemma assertion: assertion: Hence, (p4fik2 + 15(P)14klk2+2k2 ~ 24ktI:2+412-1 (piktk2 + 5~k2) I
2k t V) ~ 24ktk2+4k2 p4ktk2-2kt v: tk + p4k1k2_2klV) + p4k
24ktk2+4k2—1(p4k (p4kik2 1k2 ~ 24ktk2+4k2-1
2-
C,OROLLARY. Under COROLLARY. Underany anypositive positiveintegers integerskk,, and m m (1 (1 1 ,kk2, 2 , and (218) estimat,e (218) we we have have the estimate
~
m
~
n) for n) for the sum (224) (224)
15(P)14ktk2+2k2 (2k2)" 24k, k2+4k2 ~ (2k 2 )R 24klk2+4k2 P
4k
in(m-i) It:k2-2k —2k,+ m(m-l)
+
1 2
1
22
Nkl.n+1-m(P)Nk2,n(P)u, Nk, n+i _m(P)Nk2, ,
[ch. [Ch. II, /1, §§ is 15
WeyI's Wey!'s sums sums
114
where where
=
(em,
mm
IAjl
. . . . 0 "n ~ ,
,~n
then, obviously,
= L..J ~ N2
2 klk + 2 2 311 k2 10s2n2 2 +k 8 . S
the theorem theoremfollows: follows: Pp
Le
20n 3 20n3 ~ 2 n 2ktk2+k2
2tti /(x)
P
1 __1I_ 38k2 38k,
x=1
2n1og n 2nlogn
a2(Iog 3n-log 3n—Iog8)a)pp ~ 2 2 ee .'(log
11—
11 95n2(log 3n—log 8) 95n 2( 108 3n-log
(229) (2'29)
Note, that Note, that the thestrongest strongestestimates estimatesininTheorem Theorem 19 19 are are obtained obt,ained under underlarge largevalues values from (229) (229) that that of 8. So, for follows from 3. So, for inst,ance, instance, under under even even nn and and ss = i it follows p
Le
1
2 11'i /(.1:)
~ 11 P
1_
172n 2 •
11:=1
!) for any 8s > en we we have the estimate cstimat,e
Under an Under an arbitrary arbitrary ee (0 (0
e2n(ala:y+..• +QR~"yn)
,==1 y=I
= L...J = '""' ~1 •••• ,~"
N{P) (\ k
.
\
1\1)· • • , An
)e21f'i (Ql~Ia:+ ..• +Qn~R~R) ,
(232)
WeyPs Weyl's sums sums
12'0
summation is is extended ext-ended over over the the region region where the summation
IAII 11,' .... ,An) ~
L
,
e2 ft'i (lr\,xp:+ ..•+a,.,x,.z")
I
x=1 ~~l
~1 •••• ,~"
inequality to to the thepower power 2k 2k and and use use the theinequality inequality (141): (141): Raise this inequality
P4k2_2k( ISl 4k2 ~P4k2_2k( I
L N~P)(A1, ..... ,An) >
te21ri(lr\,x\z+,..+a,.>.,.z")
,
s=i 3: .....1
-\1 ,•.• )~,.
~ p4k -2k 2
/ (
L
2k-l
N~P) (At, ... , All)
~1 , ••• ,.\"
) 2k
P
N{P) (' k
'"
xX
) ' " e21fi (Ol.\lX+ ••. +o,,~ .. 3:n)
\
"1, ··· ,An L.-J
L.-J
.
.\1 )... ,.\"
:1:=1
(164) Hence, since by (164)
L
N~P\Ah .... ,AII) = = p2k p2k,
.\1,....-\,.
we obt,ain assertion of of the the lemma: lemma.: we obtain the assertion 4k2 ISI4k2 181
L
2 ~ p8k2_4k p8k -4k
,An)
~1,. •• ,..\"
s=i x=1
I
I
pP
2k 2k
Le L s=i I
2 11"i (Ql '\12:+".+O'R'\" X n )
~ NJ:{P)
'\1,... ,>."
L
= Nk{P) =Nk(P)
I
x=l
I
L >
P)i N~P)(l'b"',l'n)
e211'i(al~llll+ ••• +Qn~n"n).
>'1,... ,>.,.
111,···,11,.
Hence 1t follows by Lemma 1 tha,t
V~Nk{P)
L
L
N~P)(l'l'''',Jln)
I't ,.··tlln
~ [Nk{P)]2 . L
2I1a~Jlll1)'" 2IIaiiiiII) 1
min (2kP,
III ".'J,t"
~(2k)R[Nk{P)]2iI ft
L
Vi III., l 1, 1, ? 27rvQ"3
I
pp PIJ
125
Sums arising arising in in zeta-function zeta-function theory Sums
II, § 16] Ch. 1/,
then then by (239) all =
=
(-1)"-1
(-1)"-1 .
8"
+[21f-vqllt-1]2 - - -'
. = 211" vq"t- 1 = [2'7t' vq"t- 1]
>P
~
a
pn,
then by (244) (244) a
a
a
+ p']p- 2n ~ 2nP p- 2n. integers, I(x) f(x) = = alx 21. Let a1x +... 35, THEOREM 21. Let rr and and be be positive positive integers, + ... ++ anx", n ~ 35, pr ~ qq < pr+ pr+i1 , and T.,(P) 2, p 4n2, p>>4n2 TP(P) be be the number of a ~ 4n 4n 2 be be prime, prime, q = = pa, pO', pr Ta[F, P] ~ Ta[F, apB) ~ nap'- 2n = n[(a -l)p" Q
the congruence congruence solutions of the
f(v)(x)=0 (modpW), Then under under any any rr from from the the int,erval interval22e
Sz
•
p
•
, •.r=1 vIz=1
Since by (247) (247)
f(x ++p8yz) p8 yZ ) == f(x) + + /'(x)psJJz +...
+ +
1
(4 ++7)! (4r 7)!
< < p4', we
4 r+ 7 >(x)(p.t f(4r+?)(x)(psyz)4r+7 1 yz )4r+7
(mod pa), pO),
(248)
_________
WeyPs sums Weyl's
130
[Cl,. /I, [Ch. II, §§ 11
+ 7 we obt,ain obtain n1 = 4r + then setting nl •
L P p
=
e
/f'(x) ( / ' (x) t(R 1 lex) 2,rit -—yz+...+ . - ,.t+... + . ,"1 Z"l )
2'1ri
,0&-'
nll,oc-n1·
y,z=l 11,%=1
f. L..J e p
= =
,"I Z"l)
.'b1 —yz-4-...+—y"z"' 1I.t+... +~
(!.L
21rs( 21fi
'11
e
fn1
y,z=1 "z==1
where b" and qp qlJ are determined by by the the equalities equalities
1 f(")(x)
1
(ii = 1,2,...
Let as show show that that for for the the quantities quantities qu qll the the estimates estimates ps(Sr+3_P) > 22 where under under Tr = 6nl
= ni(ni + 1)
A: = nt(n; + 1) + + 6n~ = 2(4,. < 270r2, 270r2 , + 23) < 2(4r + + 7)(13r +
_ nt(nr + 1) (·.·. 1 22
€r -
.1I )'. llRl < ni(ni—1) nl(nt- 1) - n1 < 800 800
p 5 )
r,2 - r rr22 1 rr22 r2 — 1 - 4- - - - - > -
20 2k 9' 2021c>9'
4:
take place, place, then using the determination determination of the quantity quantity a, 8, we we get get 2
1512k2 < (4k)2fl1(2k)2k(2 2
< 2p
(r+1)(a.4-1) 225 104r3
Hence the the theorem theorem follows follows P
21d /(-)
Ee .=1
f
2
r(r+1)(a4-1)
< 2—
< 2p
9
by by (248): (248): P
..!.
~r.l~pISlJl+rT(P)+2P4 _ _ _0__
~ 2Pp 2P p • ·10·r'
wherei= where 7 = •. ~O,i •
r2s
!
1_i
l-~
+ rT(P) ++ 2P4 = e21r F(x,,) 5=1 y=i %=1 y=l with polynomial
F(x, y) y) + y) == alex al(x + y) +.... .. + an(x + y)". Another particularcase case of of double double exponential exponential sums with polynomial polynomial Another import,ant important particular
F(x,y) F(x, y) = = atxy +... Qnx"y" + OnX'1Y'2 + ... + was considered in §§ 16.
We We shall shall show show that that using using the repeated application of of the the mean mean value value theorem theorem itit is obt,ain ([40], ([40], Appendix II) estimates estimates for for double double exponential sums sums of of aa general general easy to obtain form P1 Pi
S(P S(P1,P2) = 1 ,P2 ) =
P2
L Le
2 '11'iF(x.,),
(254)
x=1 y=i ,=1 z=i
where "1
"2
F(x,y) = L F(x,y)=E
(255)
LQikxiyk.
j=O k=O ;=0
22. Under integers nt, n2, 2'2. Under any any positive positiveint,egers k 1, k2, k2" P1, PI, and andP2 P2 for for the the double double n2, ki, exponential sum sum (254) (254) we we bave have the the estimat,e estimate
THEOREM
p4ki k2—2k1 k2—2k2 4k k2 k ~ (2k2)n2 t k -2k p4ki p 1, p. (2k 2 )"2P!k n4ktk2-2k2N (P1 )Nk2'''2.r2 (D), S(Pi, p2 )1)14k1 (P2)u, Nk1 1 ..&2 kl,nt IS( l
2
I
2
~'
1
0',
(256)
where 1
U=
mm
. mm 1
I
..
II
),
(257)
the summation is extended k1 1,2,...... ,, nl) ni) and the extended over over the the region region IA, I'\jlI
k=O
j=1
ajt..\j
)ylI: == Po ++ /3111+. PlY + ..... + P
fl8211fl2 n2y r&2
and, therefore, 2k l IS(P1,P2)12k1 IS(P 1 ,P2)1
~pikl-l
L
N1;d(..\1, ..... ,..\nl)IEe211'i(PIY+,,,+PR2yR2)1 .
>
j=1 j=1
..\1""'..\"1
reasoning as as in in the proof proof of ofLemma Lemma25, 2, we obtain obtain the inequality inequality similar similar to to the Hence reasoning established earlier in Lemmas inequalities est,ablished Lemmas 24 and 25 25 I
S(P1,
k2 Al,...• A.,
N·(P1 )(.·. \ il
.
AI, · • ·
\ ).,. ,62tr,. i: e2 "'t mf(z) R-+oo P
L
x=1
for for any integer m
~ O. 0.
(267)
Uniform distribution Uniform d;s,trib,ution of offractional fractional parts parts
19] Chi ch. 1/1, § § 19)
143
function of of the the interval interval [0, [0,7) Let 002 ({f(x)}) -1
c=1
(269)
:1:==1
Let us us suggest suggestthat that the condition (267) isiss,atisfied. satisfied.Choose ChoosePF> Let condition (2'67) > P0(e) Po(e) in such a way way that under under nn = [~] [fl + 1 the estimate that
=
p
max
1 <m:S;;n 2
L e >2
mi(s) 2"; m/(z)
~ ""=
:&:=1
'If
4(1+2logn) 4(1 + 2 log n)
eP
(270) (2'70)
would be satisfied. satisfied. Then, using would be using the the expansion expansion of the function function "pI (({f(x)}) {f($ ) }) into into the the Fourier series, we we obtain
7—€+ m==-oo
p
00
P
-eP+ L' Cl(m)L — P = =— eP LtPl({f(z)}) Ci(m) >2 e2 I1'im / (z) >2 ({f(x)}) -1 z=l
m=-oo
:1:=1
n2
~eP+ L ' ICl(m)1 eP +>2' ICi(rn)I
z=1
m=-n2
+ L ICi(m)I IC1(m)1 + >2 Iml>n mI>n22
p Le21riml(z)
p
Le21rim/e:t)
z1 z=l
=
(the sign' sign'ininthe in = 0). (the thesmn sumindicates indicatesthe thedeletion deletion of of the the summand summand with with m 0). Hence Hence applying the estimate estimate (270) (2'70) under underJmI Iml ~ nn22 and the trivial trivial one one under under mu Iml > n2, n 2 , by applying virtue of the lemma we get virtue p
({f(x)}) — Lth({f(z)})--yP
z=1 te=l
?r
,,2
€P + 4(1 + 2 log n) eP L ~eP+ >2 4(1 + 2log n)
I
m:=-n2
2P cP + 1r2 n 2 e n2
Fractional parts parts d;s,tribution distribution Fra'ctional
144
[cIi. [Ch. Ill, III, § 19
In the same s,ame way the estimate p
:E1/J2({!(x)}) -"P ~ 2eP 2:=1
is obtained. But But then thenititfollows follows from from (269) (269) that —2eP -2eP
Np(7) — 7P ~ ~ Np('1) -1P
2eP
and, because ee can can be be as assmall smallas aswe we please, please, we we get 1
hm pI Np(-y) Np(7) lim
P—.oo
p~oo
= 7.
The sufficiency of the condition condition (267) (267) is is proved. proved. Now we shall shall prove provethe the necessity necessityofofthat thatcondition. condition.Indeed, Indeed,let letthe the function function1(3:) f(s) Now we be uniformly uniformly distributed. Take rn 0 and choose an integer q> Denote Take m i= choose an integer q > Iml. by Mk M" a set of those those x:c from from the theinterval interval[1, [1, P], P], which which satisfy s,atisfy
k+11 . ~qk— ~ {!(x)} {f(x)} < < Ie + q
((271) 271)
Denote by Tk Tk the number number of of satisfactions s,atisfactions of this inequality. inequality. Then, Then, obviously, obviously, q-1 ,-1
P
:E e2 ,..i m/(x) =:E :E e2 11'i m/(%) • k0 xEM, x=l k=O from (271) (271) that for It follows follows from for xx EE Mk Mk
{fez)}
= ~q + 6qz
I
Using Lemma 26, 26, we we obt,ain obtain Using
(mk m'. )
.mk ++2irO(k) 2'1r6(k)
2ftim!(z) = :E e2ftl. f +-,- = :E e2fl f ~ e62,rimf(x) =>
zEM"
xEM. XEMk
:cEM,
I 18
~ ~ xEM. zEA4
qq
27r1m1 fT1 2"'1~"q + =Tke 9(k)Tk, + 2wlml 9'(k)fT1 .Lk e ' .Lk, q
_
-
q
1. But then where 19'(k») l9'(k)I ~ 1. p
:E e27fi m/(x) =
+
2irImI
z=1
(272) (272)
Cit. Ill. III, Ch.
I 19J J 9]
145 146
Uniform Un,/form distribution distTibu'tion of of fractional 'rac.tional parts pa"ts
Since hypothesi,s fractional fractional parts of the function function f(x) /(3:) are areuniformly uniformly disdi. Since by by the hypothesis tributed, then 1 Til = -P+o{P). q
Take an an arbitrary arbitrary ee > 0 and choose such that that under under tJ.q = Take choose PP > P0(e) Po(e) such estimate
IJml < pi
+11 the [4-t"'1] + the
Tt - ! ~ .!EP Tk--P 2q q9 29 1
1
will and therefore there£Ol-e will hold. hold. Observing Observing that that 11 , jmJ Z ,>* z=1
,
(275) (275)
147
offractional fra,etional parts (M,rts Uniform distribution of
Ch. 11/, §§ 19] 19] ch. ill,
observing that t,hen observing then pP e2 11'i (/(s+1J)-/(X+.r»
L >
z+P
= =
:1:=1
~
L
x=x+1 :.:=%+1
2z + 2z+
P
Le
21ri (/(2:+11-%)-/(:1:»
,
x=1
we obtain from from (275) (275) we p Plpa-i1
L L
x=1 z==1
2 2
~P PI PP1
e21ri /(2:+,)
p P
+ PIS + 2 L
yO 71=0
L e ,..i 2
(/(:':+II- z )-/(x»
y>z x=1 ,>% z=l
~ P PI
+ pt + P~
p
2'11'; A I(z)
L Lee
max
l~h2 >2
,=
s';:1
1 .-=:0
P1—i 2 ' "'fP(,) 1\ -1
= Le = IIq
tr1
Ee
.
2lrt m/t
,
pI
(w+,.) .
.=:0
Since the coefficient of the highest power power z of the the polynomial polynomial /1 (JI (y + + qz) is is irrat.ional, irrational, fractional parts parts of 11 f'(y fr'actional (y++qz) qz)are areuniformly uniformly distributed di.stributed and and by by the theWeyl Weyl criterion criterion Pj -1 Pl-l
E e: >
hri -/(,+,,,) mf(y+qz)
= = o(Pl
)•
.1":=:0
But then P P
Ee >2
• '==: 1
2n mi(s) "'!(II)
Pt—i1 f . m¥(,) Pt-
= >2 E/ =
g21T1 'fl
-,-
F 1
E >2 e
2 11'i"'/1(,1+'.)
+ O(q) = o(P). = o(P). + 0(q)
.=:0
Hence, applying applying the Weyl criterion again, we get the theorem Hence, we get theorem assertion. assertion. § 2:0. Uni,form distribution distribution of offunctions functions systems syst,ems 20. Uniform and completely complet,ely uniform o,niform distribution
Let s8 ~ 1 be a fixed fixed positive positive integer, 71"" ,7. be arbitrary positive positive numbers not exceeding and/1(:1:), li(s), .•• f5(x) be be functions functionsdefined defined positive integral values exceeding 1,1,and forfor positive integral values z. x. . . . ,,1.(%) Denote by by N Np(71,. .. ,, 'Y.) the number Denote P(7l' ... number of satisfactions sati,sfaetions of the system of of inequalities inequalities
{fi(x)}
x=1,2,...,P. :a:=1,2•... ,P. {f.(x)} n. 8> Proof. Proof Let us consider the function t
mif(x + = ml/(x + mB/(x m8f(x + + s), + 1) 1)+.. +... + F(x) = 8), where mt, mj,.... . m5 are arc integers not all formula, we we obt,ain obtain where , m. all zero. Using Using Taylor's formula, t
.
f(x
+ v) =
t ~, j=O
j(j)(x)v j I
J.
" n 1 " fW(x) L vimllt F(x) = = L mll/(x ,n,,f(x ++ii) F(x) v) = = L 1- /(j)(x)
v=1 ,,=1
j=O
J.
11==1 v=1
(283)
Fractional Fra,ct;on.al parts distribution
152 152
determi.nant Since the determinant
11 11
2
1 1
2'-1
11
•.. •..
[Cli. [Ch. Ill, III, § 20
1
1
s
8
8,-1
is not equal to zero, zero, at at least least one one of of sums sums a
(j=O,1,...,s—1) (j = 0,1, ... ,8 - 1)
(284)
does not vanish vanish (otherwise (otherwise the the system system of of s8 linear linear homogeneous homogeneous equations
"
LlIjmll
=0,1, ... ,8 - 1) = ° (j(j=O,1,...,s—1)
11=1
m8 == 00 only, would contradict contradict the would have only, which which would would havethe the zero zerosolution solutionml ml== ... ... = rn, choice of the the quantities quantities m1,. choice of ml,' .. ,ma). ,m,). which the sum (284) does . D'enote least value of j, under which doe:s not equal equal zero: zero: Denote by by ti the least s
LlIimll =0
(0
~j
< t),
11=1 B
Lvtmll
(285)
=\ = A ~ o.
11':=1
Substituting these Substituting these equalities equalities into into (283), (283), we we get
= F(x)
=
(286)
j=t+1
1/=1
Hence it is seen that the the highest highest degree degree term t,erm of of the polynomial polynomial F(x) coincides coincides with the highest degree degree term term of of the the polynomial polynomial
j(t)(, ) = Ct \ n-l +... + \ t!;\ lW(s) x = nl\QnX +···+
AQt·
then under s ~ nn the function is a nonzero F(s) is Since;\ nonzero integer and and 2t ~ s3 -— 1, then function F(x) A is 2 1 with the irrational leading coefficient. But a polynomial of of degree degree nn— - t ~ 1 with the leading coefficient. But then fractional part,s parts {F(x)} are 26, and, and, therefore, therefore, the fractional are uniformly uniformly distributed by Theorem 26,
153
Uniform distrib,ution 0.(functions function:s systems sys:tems distribution of
Ch. 1/1, §§ 20] Ch. III,
+s) isis uniformly f(x+s) uniformly distributed distributed in in the the 8-dimensional a-dimensional system of functions functions f(x+l), f(x + 1),.... ,, f(x unit cube. + 1. with step step being unity: Let flOW now 83 == n + 1. Consider Consider consecutive consecutive finite differences differences with Let . .
+ 1) = = f(x + f(x + + 2)— - I(x +2) + 1), ~(2) I(x + 3)— - 2f(x + 1), + 2) 2) + + I(x f(x ++ 1), + 1) = f(x ++3) 6(1) f(x
6(n) f(x
+ 1) = fez + n + 1) -
C~f(x
+ n) +... ± C:f(x + 1).
Since transition to aa finite Since transition finite difference difference reduces reduces the degree degree of of aa polynomial polynomial by by unity, unity, 1) isis aapolynomial polynomialof ofdegree degreenn— - 1, 6.(2) f(:t + 1) is a polynomial polynomial of + 1) then 6(1) f(x + degree - 2 and, and, finally, finally, t:..(n) f(x + +1) is a constant. constant. Therefore, Therefore, with degree nn— mIl
= (-I)"C:- 1
(v == 1,2, 1,2,... (v ... ,n+ ,n+1) 1)
we we obtain obtain p
p
Le
2 11'i (ml!(x+1)+ ...+mn+l!(x+n+l»
=
x=1
Le
2 • i .6.(R)/(s+!)
= P. p
x=1
But then by by virtue virtueof ofthe themultidimensional multidimensional criterion criterion of of Weyl Weyl the system of of functions not uniformly distributed in in the s-dimensional unit cube + 1),. 1), .... ,, f(x + s) is not uniformly distributed s-dimensional unit f(x + + 8) + 1 (and, underany anyss >> n too). under (and, therefore, therefore, under too). The Thetheorem theorem isis proved proved under ss = n + completely. By Theorem 27 there 1(x) such By Theorem 27 there exist exist functions functions f(x) such that the the system 5yst,em of offunctions functions + s) under f(x + . . . ,f(x , f(x + + 1), ... under s, 8, which which does does not not exceed exceed a certain cert,ain bound, bound, is is uniformly unifonnly distributed in theorem itit is is shown shown tha,t that in the the 8-dimensional s-dimensional unit cube. In In the the following following theorem there exist functions for which the restriction restriction on on the magnitude of ss may be lifted. which the lifted. A A function f(x) f(x) isiscalled calledcompletely complet,ely uniformly uniformly distributed, clistribut,ed, if for any any ss ~ 1 the system of functions f(x+1),...,f(x+s) (287) f(x+1), ... ,f(x+8) .
is uniformly distributed in the s-dimensional unit cube. is unifonnly distributed s-dimensional unit cube. ItIt follows follows from from (280) (280) that tha,t aa function f( x) isis completely completely uniformly unifonnly distributed function 1(x) distributed if and only if under every s ~ 11 and any choice of integers integers ml, m1,.. , m8 not all zero the function choice of .... ,m.
F(x)=mif(x+1)+...+m8f(x+s) F(x) = ml/(x + 1) + ... + m,f(x + s) is uniformly uniformly distributed. is
THEOREM 28.Under Underany anyQ'a >> 44 aa function THEOREM 28. function f(x) determined determined by by the the series series 00
f(x) f(x) =
Lek=o
is completely complet,ely uniformly distributed. distribut,ed.
kQ k
x
(288)
Fra'ctional parts distribution Fractional
154
[Ch. Ill, III, § 20 20
not all all zero zero and and the function function F( F(x) Proof. Let be arbitrary integers integers not x) be be , Let ml,' m1,..... ,m. determined (288). Under 11 with determined by by the equality (288). n ~ 28 2s we we determine determine Q(x) Q(x) and and R(x) with the help of the equalities equalities n
Q(x) Q(x) =
00
L
k
L
R(x) ==
ltkX ,
k0 k==O
k ukx ,
k=n+l k=n+1
her, let Further, where uk ak = e-k . Furt Q
Qg(X) Qs(x) = miQ(x mlQ(x + +a), s), niaQ(x + + 1) + + .. . + + nlsQ(x 118(x) = R.(aJ) =
miR(x m,R(x + s). a). m1R(x + 1) 1) + + ... + + msR(x
evidently, /(x) f(x) = Q(x) + + R(x) and Then, evidently,
F(x) = ml(Q(x mi(Q(x + + 1) + R(x ... R(x++ 1») 1))++... R(x++s»)a))= =Q.(x) Q3(x) + + m.(Q(x ++s)s)++R(x +R3(x). Rs{x). + m8(Q(x in order to prove By virtue of of the the multidimensional multidimensional criterion crit,erion of Weyl Weyl in prove the the theorem theorem to show show that that under any positive integer integer sa the the estimate suffices to any fixed fixed positive it suftlces p
Le
F(s) 2 11'i F(z)
= o(P)
x=1
satisfied. Using is s,atisfled. Using Lemma Lemma 26, 2:6, we get p
Le
F(x) = =
21ti F(z)
p
Le >
2 11'i (Q.(x)+R.(:r:»
(Q.(x)+R.(x))1
x=1 p
:1:=1
~
Le
P
27riQ
,(x)
+ 21f +
LIRIR3(x)l. (x)l· s
x=1
z=l
first we shall estimate the magnitude At first magnitude of p
R ==
L IR.(x)l· x=1
Determine nn from the condition D'etermine condition
n"1
log P < (n + 1)°'
(289)
(h. III, III, §I 20] 20] Ch.
15,5 155
systems Uniform Uniform distribution of offunctions functions sys'fem'.
and choose choose PP in such such a8, way way that the the inequality inequality nn >> max(4ms, max (4m3, 2,0+1), where m= = maxl(.,(a jmvI, Im"l, isi.s satisfied. s,.ti.med,. Then we we obtain obt,ain P
•
R= R= E
•
EImyI>R(x+P) .'Elm"IER(z+lI) .1
Em"R(3:+v) ~
.=1 v1 11==1 x1
R.
p
,,=I
x=1 1:,:::=1 P
00
~s'mER(:J:+8)=8'm .:=1
:E
e- k*:E(:r;+8)" .
',=:_+1
.,=1
Hence, because because of Hence, 1 ~ (z + (% +s+ + IJ + 1)k+l - (x (% ++3)k+1 s)k+l < (P + s + 1)k+ )k ~ ~ (x + s)k
. p
LJ .=1
~LJ
k+l
.=1
follows that it follows that
R
sm ~ 8m
00
k+l'
_II"
.
i)k+1 (P ++.., 8 + 1)k+ :E : + 1 (P k=n+1
1
(290) (2,90)
•
i=M+l
Since by the determination det.ermi.nation of of n
P = pa
>
then we get for the the ratio ratioof ofsuccessive s'uccessive terms of of the t,he series series (290) (290)
(' + l)e-(k+l)Gt(P + 8 + l)k+l i)k . (k + + 2)e-'·(P + + ss + + 1).
P +8 + 1 P+s+1
< eok.. - I
P+8+1 P+ 8+ 1 P +., + 1 < 11 82n,
log P
Sn .tR)3 > > 2,
~
3
( s'n
+ 1)0' > 2tNn (n + 1)a-l >>2logP (n+1)°> 2 log P
n ( -2-
by (2'93) (293) that that it follows follows by I
q
f n+l)OI (
> /3;1 - 1 > (sn)-Se -2-
-1
> (sn)-S p2 -1 > 2P -1
~ P.
The relations (294) and (295) (295) show show that for the sum p
5S =
Le
p
2 71'i Q.(x)
= =
x=1
Le
21ri (,80+,812:+... +P,.zR)
:1:=1
the estimate obtained obtained in in Theorem Theorem 18 18 may may be be applied: applied: 3n lSI I SI ~ e p
Sinceaa >>44 and n a Since
1
1 1—.
1 2
24n2 log log n • 24n
++1)0'-1, then 1)°', then
~ log log PP < < (n (n —
1 1
2 3Jl P 24n 24n21og n ee3"P )'08 R
R
Q-'
-24 log fan < e3n-24 log
-+
0,
and, therefore, as P —, ..... co, 00, and, therefore, SS = o(P). But Butthen thenby by(289) (289) and and (291) (2'91) we we obtain the estimate p
Le
21ri F(x)
F(x)
1;=1
equivalent to to the theorem assertion. equivalent assertion.
~
181 ++ 2irIRI 21rIRI = o(P)
Fractional parts parts dis,tribution distribution Fra,ctional
158
[Ch. /II, §§ 20 20 [Ch. Ill,
distributed, then under Note. If aa function 1(x) f(3:) isis completely completely uniformly uniformly distributed, under any any choice choice positive integers integers tI and and rr the system of functions of positive
f(tx+l),...,f(tx+r) f(tx + 1), ... ,/(ta: + r)
(2'96) (296)
uniformly distributed distributed in the r-dimensional is uniformly is r-dimensional unit cube. be arbitrary integers not all Indeed, let let ml, m1,... Indeed, ... ,m all zero. To To prove prove uniform distri, m,. r be bution of the the system system of of functions functions (296) (2',96) by the multidimensional multidi.mensional criterion of of Weyl it sufficestoto show showthat that the sum suffices p
S=
Le
F(tx) , 21fi F(t:c)
= :1:=1 mrf(s ++r), where F(s) = = mtf(x mif(x + where F(z) r),has hasaanontrivial nontrivial estimate estimat,e S = o(P). + 1) + + ... ... + + mrf(x Using Lemma Lemma 2, 2, we we obtain obtain Using 8S ==
Le tP
F(z)6(5) 211'i F(:I:)c5 (x)
,
x=1 :1:=1
181 ~
~
==
1 tEE e21l'1
talzl
t
ax)
.( ,
F(:I:)+, ,
a1 x=1 8=1 z=1
-Ee211'i
a=1
tP
(F(x)+?) (F(:I:)+";) .
(297) (297)
%=1
= F(z) F(s) Determine a function function Fe(s) F.(z) by by the the equality equality F.(x) = 6. Fa (x) its its finite finite difference difference with step h: h:
+ at?
and denote by by
h
6F.(x) h
= F.(x + h) -
F.(x) = F(x
+ h) -
F(x)
ah + -. t
The difference F(xx ++h)h)— difference F( - F(s) F( x) is, obviously, obviously, aa linear combination combination of of consecutive consecutive
values of the the function function /(:1:): f(s): values of
F(x + h) m1 (f(x + F(a; + h) -— F(s) F(x) = = m1 +11++h) h) — - f(x /(x + 1») 1))+... +...
+mr(f(x+r+h)—f(x+r)) + mr(f(x + r + h) - f(x + r») = m~f(x + 1) +... + m~+h/(x + r + h), where rn'1,. . . ,,m~+h are integers not all all zero. zero. Hence, becausethe the function function f(f(s) mi, ... integers not Hence, because x) is is completely uniformly uniformly di.stributed, distributed, by (288) F(xx + (288) the function F( +h) h) — - F(s) F( x) is uniformly distributed. At differsfrom fromF(x+h)-F(x) F(x+h)—F(s) Atthe thesame s;ame time time the the function 6. Fe(s), Fa(x), which which differs di.stributed. h
by an additive constant only, distributed 88 as well. well. But But then by Theorem only, is uniformly uniformly distributed 25 25 the function F5(s) F.(x) is is uniformly uniformly distributed too. too. Therefore, Therefore, under any a from the interval 11 ~ a ~ It we have i,nterval
Ee tP
2:=1
Z,"It) = Ee
2. (F()+
11'1
tP
=. :1:=1
2 11'iFe (z)
= o(P), = o(P),
159
Normal Normal and and conjunctly conjunctlynormal normalnumbers numb,.:rs
Ch, Ill, III, §§ 21] 21] ch.
and itit follows from (2'97) (297) that follows from t
181 L
ISI
tP
Le
2 'lri F.(",)
= o(P). o(P).
a=i &=1 z=i z=l
assertion (296) (296) is is proved. proved. The assertion
§ 21. 21. Normal Normal and and conjunctly conjunctly normal normal numbers numb,ers be an an arbitrary arbitrary number number from the interval (0, 1). Let Let Let qq ~ 2 be an integer and a be by means means of of its it,s q-adic q-adic expansion expansion us write a by a = 0·1t 12 · · ·1t:
(298) (2'98)
• • • •
Denot,e number of of satisfactions of the equality . . . On) the number Denote by N(P)(Ol ... 1:1:+1 .. · 1:r:+n
(x=0,1,...,P—1), (x = 0,1, ... , P - 1),
= 01 · .• On
(299) (2'9'9)
1]and and the the equality equality (299) where Si 01 ••• arbitraryfixed fixedblock blockof ofdigits digits8,, 011 EE [0, [0, q— q-1] (2'9'9) 5,, is an arbitrary where . . . On considered as equality of of integers integers written writt,en by by means means of of their their q-adic q-adic expansion. expansion. is considered as the equality is equal, equal, evidently, evidently, to to the number of the As in §§ 8, 8, N(P)(OI .. ....8,,) on) is number of occurrences occurrences of As length n among first P blocks given block 01 ... On among the first blocks 8,, of digits of length . .
.
11 .,. 1 n , 12" ·1n +l' ... , 1p·· · "YP+n-l
digits of of the the q-adic q-adic expansion expansion (2'98) (298) for for a. a. formed by successive successive digits The number aa isiscalled called normal normal to to the thebase baseq,q, ififfor for any any fixed fixed nn equality the asymptotic equality N (P)( 01 •.• On )
~
under PP 1 under
00 -+ 00
= -qR1 p + o( P)
holds.
The theory theory of of normal normal numbers nwnbers isis closely closely connect,ed connected with problems of uniform uniform distriaq5. The general lemma lemma bution of fractional parts parts of of exponential functions functions aqx. The following following general about uniform distribution of of fractional fractionalpart,s partsofofan anarbitrary arbitraryfunction functionf(x) f(s) lies at about uniform distribution the foundation of this this connection. connection.
integers ml < LEMMA LEMMA31. 31.IfIfthere there exists exists an an infinit,e infinite sequence sequence of of positive int,egers <m2 m2 0 be given. Choose no no(€) so that for n the inequality < t is satisfied. s,atisfied. Then, Then, evidently, evidently,
n:
~ no
R
+ o(P). INp({3) - ppi ~ ~P + o{P). lNp(fl) — we obt,ain obtain Po = = PaCe) Hence Hence under under P ~ Po Po(e) we — 13PJ
and, therefore, lim pI N p({3) = P, limINP(f3)=/9,
p-+oo
which is lemma assertion. assertion. is identical with the lemma
THEOREM2'9.29.AAnum,ber numberaaisisnormal normalto to the the base base qq ifif and and only only ifif fractional fractional parts parts THEOREM the function cxqZ aq~ are are uniformly uniformly mstribut,ed. distributed.
of
Ch. Ch. Ill, III, §§ 21] 21]
Normal and conjunctly conjunctly normal norm,al numbers numbers
with 0 proof. Choose Choose an an arbitrary arbitrary block block 01 .•.• . . On of digits witll Proof. an integer uv with the the help help of of the the equality equality
161 ~
Dj
~
qq-l —1and and determine determine
0.01 . · . DR = - . 0.8i...6n. q" V
Let under a certain cert,ain xx the the equality equality
=
. . .
(301)
5,,
be fulfilled. fulfilled. Then (0
~ 8:1:
< 1),
and, therefore, the inequality inequality
v+1
v./{ aq :&:}}< -:::::: 1. Since m = = pfl .. . p~. and then dd may may be written written in form d = p~1 .•. p~., where 0 ~ k1 ~ Qt, as,..., k5 ~ Q.. a8. Note form ••• , 0 0 ~ k. Note that the the inequality inequality k,,
x0
y=i h
rn—i
P1
(y+b)z
=
m
)
Let us estimate the quantity quantity IRI. IRI. Obviously, Let rn—i
Ii
z=i
y=i
zy
Pi
rn—i
1
in z=i 2 m
Pi
.
z=O
2in
azqx
x=O
P1
4%'.
rn—i
azqz P-l """ i- . L..Je2 wm
z=i
x=O 2:=0
(325)
Let ml inj be of m m and z. be determined determined by by (318) (318) and d be be the the greatest greatest common common divisor divisor of Using applying under under dd < the estimate Using under d ~ ::. the trivial estimation and applying
::1
P1 P-l
Le
2fli
tJ%,~ azq
-,;;-
z=o
~ v'11i(1 + log m),
following from from Lemma 32, from (325) following (325) we we get
IRI~P
1
m-l
L ;+v'11i(1+1ogm)L; zi ~ p!!!:... L 1 + v'11i (1 + log m)2. (.t, m)~mml1
ml
d\rn d\m
%=1
(326) (32':6)
[Cli. [Ch. Ill, III, §§ 22 22
Fractional parts distribution
170 170
1,2,.. Since by the the hypothesis hypothesis all ~ 2 (ii (v = 1,2, ....,s), , s), then
0',
!!!.
p: = VTii·
Pl.' ·Ps ~P12 ...P88 •• •
estimate T] r1 < > 00
L 1 ~ C(e)m£, d\m
we obt,ain from (326) (326) obtain from
1+ 2 +log = ( m 1+ IRI~C(e)m2 + Vin{1+ log rn)2 m)2 =0 RI e
e ).
Now it follows from Now it follows from (324) (324) that that
T~)(b, h)= h) = ~ P + o(m~+e).
i ['In
Hence by by (322) (322) and and (323) we get get the assertion of the theorem: Hence (32,3) we theorcnl:
{tm}] P+R= qn1 P+O (1+£) m
N~)(DJ ... Dn ) = = m1 qn ++ qn
2
•
in aa part of the Note theuniformity uniformity of of distribution distribution of of digit digit blocks blocks 61 ••• Note that the .. . 6n in 32 only only ifif PP belongs follows from period from Theorem TheorelD 32 belongs to the interval interval period of the fraction ;; follows 1
E
> 22 is is a invoked for for its solution. We invoked We restrict restrictourselves ourselves totothe thecase caserim m = pO', where are chosen, chosen, as before, by (318). prime. We We assume assume that that the quantities prime. quantities r, Tj, Ti, and f3 (318). fi are
171
Distribution of 0.( digits
Ch. III, §§ 22) Ch. 22]
(a,p) = 33. Let Let (q,p) (q,p) = = 1, (a,p) = 1, a >>16(3, 16P, and r be be determined determined by the the 33. pa• If 2 we have have the the estima.te estimate equalitypr P' = pQ. equality 2 ~ rr < then we
THEOREM THEoREM
8P'
.4'·
P-l ""
LJe
211'1 --;-
,
P.
integers sand a and nn with the help Let PP> > e36 r. We determine integers help of the conditions conditions Let s
a ~ 4r
4r < s + 1,
a
n8, s > {j,
p.
< p4.,
7~n<s(p-1)
holds. by the hypothesis and pa pa = pr, In fact, since by hypothesis a:a > > 8(3r 8{3r and we obtain obtain from from (328) (328) P", then we
=
a
p. ~
s > 4r - 1 > 2{3 - 1 ~ {3, Further, evidently, n n
~
-; -— 11
aa 4r (s + 1) 4r(s+1)
m a )e21ri(1nlZt+ ...+m,x,)
(339)
1n
converges absolutely. Consider aa quadrature formula Consider formula l
i
f·· ·f f(:l:l,""
o
p
:l: a) d:l: 1 ••• dX a = ~
L f(6(k), ... ,~a(k)) - Rp[f),
k'==l
0
where -Rp[f] —Rp[f] stands stands for for the error obt,ained obtained in replacing replacing the the int,egral integral by by the the arithmetic arithmetic mean of the integrand values values calculated at the the points points (k = 1,2, ... ,Pl.
Mk =
the points are s,aid said to be nodes The set of points points Mk Mk is is called called a net, net, and the p·oint.s are nodes of the the quadrature formula. formula. Let a certain distribt ted functions functions f1(X), fi(x),. ... . . ,, /.(x) f3(x) be given. Let certain system system of of uniformly uniformly distributed 1,2,.... . . ,, s) the number Then under any Then any choice choice of quantities quantities "YII EE (0, (0, 1] 1] (v = 1,2, number of of of the inequalities fulfilment,s fulfilments of
{f3(k)) < 1. (k = 1,2, 1,2,...,P) {fi(k)} < 11, ... ...,, 00 ~ {fa(k)} (Ie ... ,P) o0 ~ {/l(k)} (340) coordinates of of the the quadrature quadrature formula is equal to 'y,P + is to 71 ;1 .. .. 1.P formula nodes nodes are + o(P). If coordinates If
determined by the equalities equalities
e1(k) = = {fl(k)}, {f1(k)), ... e1(k) . .,, e.(k) .
= {f.(k)}
(k = 1,2, 1,2,... , P), (k ... ,P),
distributed in in the the s-dimensional s-dimensionalunit unitcube. cube. In In this case case then the nodes arc are uniformly uniformly distributed by the Weyl Weyl criterion crit,erion the equality equality p
Le
2 11'i (mlet(k)+... +fR.e. (k»
k:=l
== o(P) o(P)
(341)
177
Conn,ection formulas Connection with quadrature formulas
Ch. III, Ill, § 23] Ch.
,m.
of integers integers ml, m1,.. choice of .... , m8 not all zero. zero. We shall denote the sums holds under any choice (341)by byS(ml, S(mi,... (341) ... ,m 8 ): p
S(mi, . SCm 1,···, .
.
~e21ri(ml(1(1t)+ •••+m,e.(k» m B ) --= L..J
k=!
formula. sums corresponding corresponding t,o to the the net net of the quadra.ture quadrature formula. and call exponential sums THEOREM s,eries of of a function /(X1, . ,, XB) converge absolutely, THEOREM35. 35.Let Let the Fourier series andS(ml, S(mi,... ,rn3) ,m8) be C(mt, be its Fourier coefficients coefficients and ,m be be exponential exponential sums sums CY(mi,.... .. ,m,) corresponding of a quadrature quadra,ture formula formula corresponding t,o to the net of .
,l )
J...J
I I p
= ~Lf(el(k), ... ,e.(k»
f(ZI, ... ,:l:.)dz1 ••• dz.
o
-Rp[f]·
1t=1
0
Then the the equality* equality· 00
1 Rp[f] Rp[f J = = p-
L'
(mj,.... . C(mt, ....,m3)S ,m,.)S (mt, ,,m3) mB) .
.
(342) (342)
ml •... ,m.=-oo
holds and the the error error Rp[f] Rp[f]tends t,endstotozero zeroasasPP—+ -+ 00, oo, if and only if the nodes of the quadra,tul~ formula are are uniformly umformly distributed distribut,ed in in the the s.-dimensional s-dimensional unit cube. quadrature formula
Proof. Since Since
J...J t
C(O, ... ,0) =
t
f(ZI, ... ,z.)dz t
o
.•• dz.,
0
using the the expansion expansionofoff f(zi,. then using (x 1 , ••• X II) in the Fourier Fourier series series we we get .. , x3) p
Rp[f] = Rp[j] = ~ Lf(6(k), ... ,e.(k» -
- 0J..
. .
k=!
1
=p =
P
L
L 00
J...J 1
1
.. ,x.)dzt ....dx8 f(zt, ... dz.
.
. .
0
(k)) - C(O, . .. ,0). (mjtj (k)+...+m.t. C(m}, ...+m.Mk» C(mi,.. .. ,, m.)e2"'i(ml~1(k)+ — C(o,. .. , 0).
k=l ml,...,m.=—oo f'lJ.t ..... m,=-oo k=1
=
Hence after after singling singlingout out the the summand suminandwith with(ml, (mi,.. ... ,m.) Hence (0, ... ,m8) = (0,. , 0) and changing . . ,0) the order of summation we have the equality .
1 Rp[f] =p
L' mt ,...•m, ::::-00
'Henceforward
E
p
00
e2 1t'i (mlel(k)+ ... +m,(.(k» C(mi,. C(m m.) 1,···,,m8) II L.J ' . .
~
k::1 k1
, signifies that the summation is over s-tuples (ml,
, 0). m8) :1= (0,. signifies that the summation is over s-tuples (ml, ... (0, .... ,0). .. . , m,)
Fractional parts parts distribution Fractional
178
(Ch. III, § 23 [Ch.
coincides with with the the first first assertion of the the theorem theorem by by the definition which coincides definition of the sums sums S(ml, ,m.. ). S(mi,..... . ,rn8). Now we turn turn to the proof assertion. Let Let the nodes nodes of of the the quadrature Now we proof of the second second assertion. formula be uniformly formula uniformly distributed in the s-dimensional s-dimensional unit cube. Then Then by by (341) (341) p
S(ml, ... ,m9) ,m,)= =
:Le21ri(tnl(t(k)+...+m,e.(k» =o(P). = o(P).
(343) (343)
k=1
Take an arbitrary arbitrary ee >>00and andchoose choose mo mo
mo(e) and Po so that that the estimates = mo(e) Po = Po(e) so
Ll= L IC(mll,,,lma)IIS(mll,,,,mS)I""z.») ...
o
Ox1 . . . Ox, Xl ••• 3:.
2
d·Xl
• ••
dx.
(349)
/c
~
0
0
satisfied and and its partial are satisfied partial derivatives derivatives
an I(xl, x,,) (xi,...... ,,x,) ,,• 8X~1 ••• ax:' •
•
.
continuous with respect to variables are continuous variables with nj = s,atisfy the Dirichiet Dirichlet condicondi= 0 and satisfy tions with respect to to other other variables. varia.bles.
THEOREM 36.Let Let/(:£1"'" f(xi,. .. , x,) W,(C) and THEOREM 36. x.) be be an an arbitrazy arbitrary function function from from the class class W.(C) and .. ,,1s) be determined by the equality (348) the quantity Rp(11, ... det,ermine,d by (348) constituted constituted for for cocoordinates ordinat,es of of the net of of quadrature quadra,ture formula formula
1..·1
I I p
/(ZI, ... ,Z.)dX l ... dx.
o
= ~ L /(6(k), ... ,e.(k» -
Rp[/l·
k=l
0
Then for the error error of ofthe the formula formula (350) (3,50) we we have the relations rela.tions
1)81 I 1
R p [fl -( - -
1
· .. .
o
0
08
/(x 1 ,".'x.) 0' fJ Rp (Xl,···, X" ) dXl ... dx", Xl •••
Xs
IRp[/li ~ ~T(P), IRpEfil where where T(P) T(P) isisthe themean meansquare s,quarediscrepancy discrepancy of of the the net. net.
(350)
12 18'2
Fractional Fra,etional parts distribution distribution
[ch. [Ch. Ill, III,§§23 23
Using the the first first of ofthe theconditions conditions (349), (349), we we obtain obtain Proof. Using
8 11 - 1 f
x8) a) (Xl,' •• ,:£,,-1, 1, 1, X,,+l, • ..•• ,X
8-X l ..•••. 0X1
83:,,-1
=0 =
( = 1,2,. . . ,3).) (ii v = 1, 2, ... ,S •
then, obviously, obviously, But then, 1
8"f(X1, ... ,X",e"+1(k),.",e.(k)) dx" f &f(xi,... 8X1 •. , 8x" /J .
.
(,,(k)
1 = = &''f(xi,.. 8"- f(X1'" .,x",e"+1(k), ... ,e.(k» /%."=.1 .
Oxj aX 1 .• .• •. OXv_1 8X,,-1
x":=E',, (k)
.. , es( k) ) 81/ -1 f (XI, ••• , X ,,-1 , ell(k), ... &_1f(si,..
—
=
Ox1 .••. •8-X . O-Xl II - l
—
'
and, therefore, I
11
1
/ ...e./ I
I
J
8 S j(x 1 , ••• ,xB )d d dx1...dx3 Xl··· X. 8Xl ••• 8 X s Oxi...0x3
J
~1 (k)
(k)
I
1
j ··· J/ ...
=-
~I (k)
as-I/(Xl, ... ,XS-l,e,(k))d f) . . 0x8_j {) Xl Ox1 Xl ••• X.-l
J
/
.
• ••
d dX ..-l
= ···
~. -I (I~)
= (-1)8 f(el(k), ..... , es(k)). = FUrther observing that Further 1
{)"-1 f
d _— 8" f(xl"'" x.. ) xvxv—xp I— {) XII XII - XII {}
/J o
Xl ...
(351) (3.51)
r,) (Xli" .,x.) 8
aXI ...
XII
X,,-l
1
/1 _ / 0o
I
j
0
{)"-1 f(X1"'"
{)
x.) dxi, XII
Ox1 ... . . . OXv_i OXl XII -l
1
-— _ / 8"-1 f(X1" ",x.) dxv, -——I aOx1 ,x", .. . {)X"-l Xl •••
j
o0
we get get 1
1 fO$1 88/(:£I,""X,,)
··· /j a.0x1...&c8 {J . x, j1...
/ o0
d
. Xl'" Z. $1···
Xl •••
d x.
00
1
=-
/ o0
1.
··· /
1
EP- /(Xl,""X.)
0-Ox1 {) Xl ••• X ..- l . . .
00 1·
dx 1 • •• x.
= ···
1
= (-It / ... / f(:l:lI"
o
d
Xl • • • 3:,.-1
0
.,x.) dx 1 ••• dx..
(352) (352)
183 183
Connection with quadrature formulas Connection formulas
Ch. III, Ill, §§ 23] Ch.
We det·ermine determine a function t/J(x, y) y) with the aid of the equality We il/,(X 'Y
)_ { 1
0
,y -
if 1, if xx < ~ ,.
(353) (353) ,---
e.(k»
(k = 1,2, 1,2,.... .. ,, P) lying in (Ie
for the number of the net points Then for point,s M(e1(k), M(el(k), ..... ,, xi < 11, ... ...,, 00 ~ Xx8s < IS we get the region 0 ~ Xl p
Np(7i,.. ., III) = Np(11'.'
L 1fJ(el(k), 11)' . . 1/J(es(k), I.) 1:=1
and by and by (348) (348)
1 p
Rp(7i,...,73) Rp("Y1,'" ,7s) = P L ¢(6(k),7d·· '¢(~s(k),7s) -71 .. ·7s·
(354) (354)
k=1
Using the equalities equalities (a51) (352), we quadrature (351) and and (352), we write write down down the the error error of of the quadrature formula in in the form formula p
Rp[f] Rp[J)
1
1
...dx8s = ~ L!cel(k),.oo,~s(k)) - I / ... / !(Xb oo "xsdx1 )dx 1 °o'dx ...If(xi,...,xa) k~l 0 0
-- 2e
,,'
P2
. .
k=l
coincideswith with the the sum sum (a62) (362) and therefore IT m,,) coincides then the sum sum S(mt, S(mi,.... .. ,,m3) If PP = p2, then under(ml'." (mi... ,m9,p) under ,m",p) = 11 the estimate (s — IS(mt, (8 -1)p -1)VP IS(mi,.... ,m.)1 ~ (s — l)p = (8 . .
(363)
The following following theorem is based on the use of this estimate. estimate. is is valid valid for for it. The , x9) belongs THEoREM37. 37.IfIfa.afunction functionI(Xt, f(xi,.... .. ,x,) belongs t,o to the class E~(C), pp is a prime THEOREM prime grea,ter quadra,ture formula then for for the error of the quadrature greater than than s,a, and and PP = p2, then
we we have have the estima,t,e estimate
s,s, (ap,p) ,X (a",p) = 11 (I) 1,2, ... ,8), and 8 ) E E:(C), p> P= =p.p. For For the theerror errorof ofquadrature quadra,tureformulas formulas with with parallelepipedal parallelepipedal nets
THEOREM38.38.Let Let/(3:1, f(xj,.... .. THEOREM 1
1
f·· ·f I(xl, ... , x,) dXl · · · dx, o
0
(369) (3-69)
parts distribution Fractional parts
190
[Ch. III, §§ 24 24 [Ch. Ill,
we have the estimate
+... + a3m3)
lRpEf]l
(370)
be chosen chosenso sothat that under under any any aa < 1 the net of of the the formula formula (369) (369) can be Rp[f) ~ eel CC1 IRp[fJl
logas P
pa
'
on aa and s. where 0 Ci(ct, s) is a constant canst,ant only only depending depending on C11 = Cl(a, that for Proof. ItItfollows follows from from the the definition definition of the class class E~(O), that for the theFourier Fouriercoeffico,efR, x8) cient,s cients of ofthe thefunction functionf(xl' f(xi,.... ,x B ) the estimate , IC(,ni, le(ml, .... ,m.)1 ~ (-
C c
- )a ml···m" )
holds. then by by Theorem Theorem 35 35 holds. But then 00
L:'
IRp[f]I
IS(mi,
IC(mi,.
. ,
I
..
get the first first assertion of the theorem: Hence, using using the equality (368), (3-68), we we get 00
e",,' Rp[f] I ~ P LJ IRp[f]l
I
L__l
=—oo Jnt,.·o,Jn,=-oo 00 00
6p(aimi+...+aama) o,(alffil +... + a"m,,)
",,'
—c =0 —
LJ
m .. (-mt··· -)a
Jnl.ooo,m,=-oo
(371)
To prove the second with the aid of the second assertion assertion we we determine determine integers int,egers P1 PI and P2 with equalitiesPIp' = [~], P2 = [i] and replace mIl by by n"p npp + mIl in (371): equalities
C fll,..fl.=—C0, nl. ,n,=-oo, -PI ~ml.o..• m, ~P2
o,(alml + ... + + a.m.) (nIP+ml" .n"p+m,,)a·
o ••
Since, obviously, obviously, under under m E [—P1, Since, [-pI,P2l P2] 00
00
(np + m)°'
+2
(np
—
1, then then are chosen > 1, P2 P2
+ ... + a3m3)
/
L:' ,m1
—pi
(
a
P2
+
... + a3m3)\ — —T a (ai,...,a8). —
m1...m3
p
using the estimate (376), obtain Therefore, using (376), we we obtain
+...+a3m3) < Substituting this this estimate into into (373), (373), we we get get the the second second assertion assertion of ofthe thetheorem theoreln under a certain C1 C 1 < (:~~) Q,:
3a \8 1+2a(3+2logp)as
[I] I ~ (~). C 2 tog P)a. ~ CC ( C 1 + ~(3 + IRRp[f] p ~ a - 1 1P ~
1i
logasP toga. p = CC toga. p . pa pOt
—
t1
pa pOt
If there exists exist,s an an infinite infinit·e sequence sequence of of positive positive integers integersppsuch suchthat that under under certain = al(p), ai(p), ..., C0 = pes), a(s), and a1 Co = Co(s), fi P= at = ... , a3 as = a8(p) a,,(p) the estimate
'2
6,(aim1+...+a3m8)
L:' =
p
(377)
=
[nj'] and to the sequence, where [l.jl] andp2 P2 [f], holds, then for every every pp belonging belonging to sequence, where PI p' the integers integers at, a1,...... ,,as mod uloppand and the the nets a8 are called optimal coefficients coei1icients modulo
Mk = M( {a~k },...,{a;k})
(k=1,2,...,p), (k = 1,2, ... ,p),
corresponding to to them, them, are s,aid said to be optimal parallelepipcdal corresponding paralle1epipedal nets. It is seen from Theorem 38 that optimal parallelepipedal nets enable enable us us to to construct It from Theorem 38 that paralle1epipedal nets quadrature formulas, for the error of which the estimate formulas, error of which
Rp[fl
= OCO~:P)
(7=03)
(378)
holds. nets it is impossible impossibleto to obtain obtain the error holds. It It can can be be shown shown that thatfor for any any choice choice of nets term better than term If p[f] =
P)
(379)
Ch. 24] Ch. III, §§ 24]
Quadrature interpolation formulas Quadratureand and interpolation formulas
193 193
on classes E~(C). Thus the the estimate estimate (378) (378) is is close close to the best best possible pos:sible in in principal principal on
order and and only only the thelogarithmic logarithmicfactor factorcan canbe beimproved. improved. Let us note some other characteristic peculiarities of quadrature formulas with parallelepipedal net,s. nets. It is allelepipedal is seen seen from the estimate estimate (378) (378) that that such such quadrature quadratureformulas formulas react automatically automatic,ally to the the smoothness smoothness of the integrand: int,egrand: the the smoother smoother the the periodic periodic results are are ensured ensured by by the application ... ,, x.») precise results application of function I(xl, x8), the more precise function f(xi,... one same quadrature quadrature formula. formula. This Thisproperty propertyof ofcomputational comput,ational algorithms algorithms one and the same (see [2]) [2]) isis called called their their "insa,tiableness". "insatiableness". Thus the quadrature (see quadrature formulas formulas with with paralparallelepipedal nets enjoy enjoy the the property propertyofofinsatiableness. insatiableness. solutions of minimal value value of the product ml ... m, for nontrivial nontrivial80lutions Denote by q the minimal the congruence (380) atml + +... a.m, == 00 (mod p). pl. ... + + a,m3 . . .
with parallelepipedal parallelepipedalnets netsisisthe thefact factthat that Another peculiarity of quadrature formulas formulas with they are exact exact for for trigonometric trigonometric polynomials polynomials of the form Q(Xl," = Q(xj,. .,x,,) ,x8) = . .
L >
2 1 t . C(ml," C(rni,. .,m 8 )e 21'i(m x +... +m,x,), .
(381)
mt ...m, 22 the Proof. A (x 1 , •.•. , xx3) s) implies the the existence existence and and continuity continuity of of the derivatives belongs to to the class E~( G), implies belongs derivatives ""' ) I Tt ,.••• '1'. (~..... 1,· · · ,x,) '''''.t
= 0,1, vii = 1,2,... (TV =0,1, 1,2, ... ,8).
.
Let s = = 1, a> a >2,2,and andf(x) f(x)E EEr(C). Ef(C).Performing Performingthe theintegration integration by by parts parts and and using the periodicity periodicity of the integrands, int,egrands" we obtain
/
f'(y)
—
x}
dy —
=
f
f'(x + y) —
and, therefore,
=
/
f(y)dy+
dv
-
=
x} -
>
weget get the the lemma Applying this this equality the variables variables Xi,. Xl, ••• X s consecutively, consecutively, we equality to the .. ,, x8 assertion:
f(xi,.
. .
,x8)
I frl(yi,X2,.
=
— Xi}
.
—
1)nldy =
= o
o
Note. integer, aa > r + Not,e. [f If r is is a positive integer, we have the the following following + 1, and fI EE E:< C), then we equality analogous to the the equality equality (387): (387):
f(x1,. . . ,x8) I(xl,"" x,,) = =
I /J.1 ... /1, [< l)r-1 ] xv))] L_ r T1 .....TT·(Yl, ....,ya) ,Y.)]J II -r! BT({y.,-x.,}) II
—
'1'1, •••• '1',-0
0o
.
0o
11--1
'1'.,
dyi ... .. dy., dYl
Fractional parts parts distribution Fractional
198
where B r ( x) are the the Bernoulli B,ernoulli polynomials: polynomials: 1·
B 1 (x)
= X - 2'
1
2
B 2 (x)_=
X
[Ch. ",, Ill, § 24 [Ch.
-
+ 6'
X
Under r = 11 this assertion assertion coincides coincides with (387), and in the the general general case case it it is is proved proved by induction with respect respect to to rr with withthe theuse useof ofthe theequalities equalities
Br(1)
= rBri(x)
Br(O),
(r
~
2).
THEOREM 39. Let Let r ~ 2 be a positive integer,Qa ~ ? 22 r, and . ,as THEOREM 39. positive integeJ:, and a1,. al, ... , a3 be optimal x8) belongs belongs to to the class coefficients modulo class E~(C), then coefficients mod ulop.p.IfIfaa function functionf(Xl,". f(xi,. .. ,,x,,) we have the equality we have .
. ,x8) f(Xl,."'X.) .
=~
t t _rrt..... k~l
a~k },... ,{a;k}) ~ [( _~r-l B {a;k r (
rtt ... ,r.-O
Xu } )
11-1
p)
r
})JT&.p
8
rr •( {
0(1ogYp" + o (log"Yr ' + \ p' /
(388)
where where aa constant 'y 'Y depends depends on r and andasonly. only.
Proof. Let f1(x1 ,...,x8) and and f2(Xt, f2(xi,..... ,x,,) ,x3) ixiong Proof. Let functions functions 11(Xl,".'X,,) belong to to the the classes clas:ses E~(Cl) respectively. We We shall shall show showthat that the product of and E~(C2), reS'pectively. of these these functions .
f3(x1,...,x3)= 13 (:VI , •••, Xa ) = 11 (Xl' · · · , Xa )!2(XI, · ·..,x8) · , X.,) belongs c33 depends a, and belongs to the class class E~(C3), where where C depends on Ci, C1 , C2, C2 , Ol, and a. s. Indeed, denote by by Cj(ml, C,(rni,.... . , m3) = 1,2,3) the Fourier coefficients of the func) Indeed, denote ,m (j = 1,2,3) the Fourier coefficients funca Multiplying the Fourier series of the functions fi and 12, and 13. 13' Multiplying the Fourier series of the functions 11 12, we we tions 11, tions fl, 12, and obtain .
00
f3(xi,.
. .
x.)
(m .. E CC3(mj,. = mt •.··.m.=-oo 3
m3) 1,· · • ,, m •)
e2'ft"i(ml:r;1+ •.• +m~x.) ,
where 00
C3(mi,. .
.
=
E
fl,lJ ••••
.. ,n8) C2(mi — ni,. . . ,m8
—
n3).
n.=-oo
Therefore, 00
E
JC1(ni,...,n3) C2(ml —ni,...,m8
nI,...,nI=—co 1lt,.·.,n.=-oo 00
[n1
.. .ñ8(mi
—
ni) ... (m8 — (389)
Ill. SI 24J CIt. III, 24]
Qua,d,.tule and interpolation intetp,oIation formulas form,ulas Quadrature
191
where a(m) denotes the sum s'um
Estimate the sum u(m). a(rn). If iii> m >1, 1, then then
InI>—In,J
>2
[n(rn-n)r
>2 1
11= —00
estimate is, = 1 too. But This estimat.e is, evidently, evidently, satisfied s,atisfied under m = But then then we we get get from from (389) (389)
and,
=
,m9)I
ml •..·,nl,=-oo
where (21ri)r(r1 +... +r,). Since Since where 0' C' == (2 rTI rr O( )1 1 1 ... m B' mI,···,m.. ~
I
I c Imt(-
ITt
IrT'
B · · .m -)01 ml···m. ~ C ~ C -. .: : (()r' m5)r mI .. . -m, )OI-r -. .: : (ml mt .. . -m,
fi
,y,) belongs to the the class class E~(Cl) E(C1) with the function function ft(Yl," fi(yi,.. .,Y.) = = jrr1J •.•• rr'(Yl,""Y') belongs to with
the constant C1 = IC'IC. 01 = JC,IC.
ofthe the r-th r-th Bernoulli Bernoulli polynomial polynomial BBr({y}). Let c(m) be the the Fourier Fourier coefficients coefficients of Since r ( {y }). Since for the Fourier of the the function c( m) == Fourier coefficients coefficients of c(m) = 0 ( ~r ), then for
12(Yl, f2(yl,.... ' ,Y.) = =
..
Xv)) II B~"({y" -—x,,})
,.,=1
we obtain the estimate we obtain
C2(mi,...,m9)=O( — (in1
.
m8) r)
, belongstotothe theclass classE:( E(C2). But then the and, therefore, the function f2(Yl, .•• ,1/,) belongs O2 ). But determined by by the the equality (390) belongs belongs to to a certain function equality (390) certain class clas:s function F(y!, FQji,...... ,, 1/8) det,ermlned may use the E~(C3) and for the the evaluation evaluation of the integrals integrals in the equality equality (391) (391) we may obtained under under PP = = p in Theorem 38: quadrature formula formula obtained 38:
J...J /.. / F(yi,... 1
1
dy9 , y3)dy1 .. . dys F(yt, ... ,ys)dYl'
o
.
0
depends on rrand and s8 only. where'Y only. Hence Hence by by (390) (390) we have have the equality where depends
Ch. Ch. Ill, III, §§ 24]
201
Quadrature and interpolation formulas Quadrature and interpolation formula:$
coincideswith with the the theorem assertion by by the definition theorem assertion definition of the the function function which coincides
F(yi,.. F(Yl, ... ,Vs). , y.). .
is obtained obtained under under the the assumption assumption that that the The interpolation interpolation formula (388) (388) is the function function belongs to the class f(zi,. f(Xl"'". ,x8) XII) belongs class E:(C), where a ~ 2r and r ~ 2. In In the the same sameway, way, somewhat ourselvesof of the the validity validity of of the somewhat complicating complicating the proof, proof, we can convince convince ourselves also. So if /(3:1, f(x1,... and al, a1,.. formula under underrr = 11 also. formula So if ... ,x3) ,3: 8 ) E E~(C) and ....,a, ,a., are are optimal optimal coefficientsmodulo modulop,p,then thenunder under PP = p we have the equality coefficients we have .
f(xj,.
.
.
ft
=
+
—
o (log"Y
P)
(392)
p'
where I depends depends on on s only. only. Unlike Unlike the the formula formula (388), (388), which which is not not unimprovable, unimprovable, order of of the error error decrease decrease in the interpolation interpolation formula formula (392) cannot be improved the order under any any choice choice of nets. The quadrature quadrature and andinterpolation interpolationformulas formulas with with parallelepipedal parallelepipedal nets net,s established established section were wereobtained obtainedunder under the the assumption assumptionofofthe theequality equalityPP = = p, where in this section where P is is If the the the number of the net nodes nodes and and pp is is the the modulus modulus of the optimal coefficients. coefficients. If quantitiesat, aj,... a8 are chosen quantities ... , as chosen so that that the thenumbers numbers1,1,a1, at, ..... ,, a8 as are (s + +1)-dimen1)-dimensiona! too, sional optimal optimal co,efficient.s coefficientsmodulo modulop,p,then thenthese theseformulas formulasare arevalid validunder underPP
