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n, (at , . . . , an , p) al X + . . + a n x n . Then the following estimate holds:
=
1, and
f(x)
=
.
pQ 2"i f( x) L e pQ x=l
� (n _ l )pa - l .
y and z run through complete residue sets modulo pa - l and p, respec tively. Then the sum y + pa-l Z runs through a complete residue set modulo pa and, since 0' 2 and p > 2, we have
Proof. Let
;;;::
[ Ch. I, § 4
Complete exponential sums
26
Therefore,
pQ e2 11'i pQ
f(x )
L
x=l
pQ - I
2 11' i f(y+ p Q - I z)
P
L Le y=I z= l f' (y) z pQ""'- I 2 11'1. f ( y) ""' 211'1. -= w e Q we p z=l y= l (y) f . 2 11'1 =p L e y= l
pQ
=
P
p
P
Q-l
pQ 8p [f' ( y)] .
(70 )
But then f(x)
pQ e pQ L 2 11' i
�p
x=l
pLQ - I 8 [J' (y)] = pOl - l 8 [J' (y)] = pOI-I T, L p p P
y= l
y= l
(7 1 )
where T is the number of solutions of the congruence
f '( y) == 0
( mod
p).
Since (a l , ' . . , a n , p) = 1 and p is a prime > n, then at least one of the coefficients of the polynomial f '( y) = a l + 2a2Y + . . . + na ny n - l is prime to p and, therefore, T � n - 1 . Substituting this estimate into ( 71) , we obtain the lemma assertion. It is easy to improve this result for polynomials of a special form. Let p b'� prime, (a, p) 1, and pQ 2 11'i �
=
S(a, pOl) = L e x=l
Let's show that under
a
� 2 and
pQ
n � 3 the following equalities hold:
S(a, pOl ) = { �:=�S ( a , pOl -n )
if 2 � a � n and if a � n + 1 .
Indeed, from (70) it follows that
l 2 11'i p p QL S( a, pOI ) = P e Q 8p ( nay n -l ) . a y"
y= l
Hence under
(n, p) = 1 we get
the first equality of ( 72 ) :
(n, p) = 1 ,
(72)
Ch. I, § 4)
Simplest complete sums
27
be the greatest power of which divides Then, Now let a � + 1 and +1 � - 1)(3 + 2 and considering separately the cases using the estimate a � a � 2 (3 + 2 and + 1 � a � 2(3 + 1 we obtain
p, n pfJ pfJ (p n (y pcx - fJ -lzt y n npa - fJ-l y n -lz (mod pa ). +
==
n.
+
Therefore ,
y=1 -fl l 11";%=1ayB a2:- 2 p p fJ 1 = p + y= l e Q Dpp + l (anyn - l ). Hence, since (an,pfJ+ 1 ) = pfJ, we have pQ2:-fl-1 2 11" ; ayBp 1 S( a, pCX) pfJ+ y=l e Q Dp( y) a-fl-2 p 2 11"; paya -B 2: +1 pfJ y=1 e B = pn -1 S(a,pcx - n ). =
=
Thus the assertion (72) is proved in full. THEOREM 6 . Let sum
the estimate
n and q be arbitrary positive integers and ( a, q) = 1 . Then for the S(a,q) = :c=2:q 1 e2 1I"'' -qa:c-B (73
holds.
Proof. Since 1 , then under 1 and Lemma 2 and Theorem 3 , respectively:
(a, q) =
n=
2:q e211"; �q = 0, :c=1 Therefore it suffices to consider the case
)
n 2 the estimate (73) follows from 2:q e2 1I" ' -axq-2 � y'2q . :c=1
n � 3.
=
[ Ch. I, §
Complete exponential sums
28
At first we shall show that for any prime p under
a
� 1,
4
n � 3, and (a, p) = 1 ( 74 )
where
Cp (n)
Indeed, under
Let 2 �
a
if p < n6 , if p > n6 •
1 by (60)
a =
� n and
= {�
=
(n, p) p .
if if Then p �
p < n6 , p > n6 .
n and, using the trivial estimate, we get
IS( a , pa ) 1 � pa � pa ( 1 - � ) p � npa ( 1 -� ) . Let finally 2 � a � n and (n, p ) 1 . Then by (72)
=
=
IS(a, pa ) 1 = pa - 1 pa ( 1 - !; ) � pa ( 1 - � ) . ( 74) is satisfied under 1 � a � n . Apply
Thus the estimate the induction. Let under 1 + (k - 1 )n � a � kn this estimate be valid for a certain k � 1. We shall show that the estimate is valid under 1 + kn � a � (k + 1)n as well'. Since it is plain that 1 + ( k 1 )n � a n � 1 + kn then using the equality (72), ' by the induction hypothesis we get -
-
Thus the estimate (74) is proved for any a � 1 . Let now q = pr1 p�. be the prime factorization of q. Using the multiplication formula (34), we obtain •
•
•
n q ""' 211'1 axq L.J e .
-
Then, obviously,
p� .
""'
L.J e
2 11'i
ab.x n p". •
(75)
,,=1 x.=1 are coprime with p" . We determine a" with the help of the x=1
where the quantities b " equalities
=
II8 (v
( a " , p,,) = 1 and by (75)
=
=
1 , 2 , . . . , s).
S(a, q) S( a 1 , prl ) . . . S( a8 ' p�· ) .
Ch. /, § 5)
Mordell's method
29
Hence, using the estimate (74) and observing that the number of primes less than n6 does no t exceed n6 , we get the theorem assertion: I S(a, q) 1 �
Note that under n
>
al ( l - � ) a . ( l - � ) n 6 1 _ 1n . . . . Cp . (n)p s �n q Cp 1 (n)P 1 2 , q = pn and (a, p) = 1 by (72) for any prime p pn 2 .,.. i axn '" e p n = pn - 1 = pn ( 1--n1 ) . L...J x=l
Therefore, in this case
1 = q _ 1n .
S(a, q)
Thus under fixed n and increasing q the order of the estimate (73) can not be im proved. Let f(x) = a1 X + . . . + a n x n , (a1 , ' " , a n , q) = 1 and Seq) be a complete rational exponential sum of the general form Se q )
q
= Le
2
. In
f(qx)
-
.
(76)
C(n)l - n ,
(77)
x=l
In Theorem 6 the estimate I S(q) 1 �
1
where C(n) = nn 6 , was proved for polynomials of the special form f (x) = a n x n . With the help of the significant complication of the proof technique, Hua Loo-Keng showed that under certain C(n) the estimate (77) is valid for arbitrary complete rational sums (77) as well. A proof of an estimate close to (77) can be found in [16] and [44] . § 5. Mordell's method Let us consider a complete exponential sum with a prime denominator S(p)
P
= Le
2 1r i
al x+...+an x
n
p
x=l
Mordell [36] proposed a method of such sums estimation based on the use of prop erties of the system of congruences . n . .. .. . x f + . . + x : == y f + . . . + y :
�� �. : .. �.�.� � ��..�.... : .. � � . .
.
.
.
.
}
(mod p),
( 78 )
Complete exponential sums
30
[ Ch. I, § 5
where p is a prime greater than n and the variables X l , , Yn run through complete residue sets modulo p independently. First of all we shall prove a lemma about the number of solutions of a congruence system of a more general form. •
•
LEMMA 5 . Let q1 , . . . ; qn be arbitrary positive integers, q be the number of solu tions of the system of congruences
��. .�. : . '. ".�.�.�..�.�� .�.:.'. ".�.�.� ....�:��. .��.� . x f + . . . + x� == y f + . + Y k (mod qn ) 71"1 ( Le .
.
Then
.
9
2
.
•
=
LCM ( q1 , " . , q n ) an d Tk
(79)
}, a1 x + + an xn ... 91 9n
) 2k
x=l
Proof. Since the product
equals unity, if numbers X l , otherwise, then, obviously,
•
L q
Tk =
•
•
, Yk satisfy the congruence system (79) , and vanishes
.
8q 1 ( X l + . - Y k ) .
X l , . . · , lIk = l
. . . Oq" ( x f + . . . - Yk ) '
Hence, using Lemma 2, we get the assertion of Lemma 5:
Tk = q1
1 . qn
-- .
.
q
Le
2 71"1.
(
a1 x
91
+...+
an x qn
n
) 2k
x= l
In particular, under
Ie
=
n and q 1 = . . . = qn = P ·it follows from Lemma' 5, that
1
Tn == -;
P
where Tn
is
P
P
L L
4 l , • • • ,a n =1
%=1
e
2 ' a 1 x+ ...+a n x
71" 1
the number of solutions of the system
P
(78).
n
2n (80)
LEMMA 6. Under any n � 1 and a prime p > n , the number of solutions of the system (78) satisfies inequality Tn � n ! pn .
Ch. /, § 5]
Mordeli's method
31
Proof. Let .A 1 , . . . , .A n be fixed integers, 0 � .AI' � P - 1 , and let T ( .A1 , " " .A n ) be the number of solutions of the system of congruences
��. �.:.'. :.�.�.� .�. .�.
x
We shall show that
1
.
.
f + . . . + x : == .A n
1 � Xv � p.
(mod p)
}
(81)
(82) Indeed, we introduce the following notation for the elementary symmetric functions nd a the sums of powers of quantities X l , . . . , X n : 0' 1 X l + . . . + X n , . . . , O' n = X l . . . X n , 81 = X1 + " , + Xn , . . . , 8n = xf + . . . + x: . =
Let X l , . . . , X n be an arbitrary solution of the system (81). Then, obviously, 81
==
.AI , " " 8 n == .A n
and using the Newton recurrence formula vO'v = 8 1 0'1' -1 - 8 2 0'1' -2 +
under v
=
1 , 2, . . . , n we have vO'v
==
.
(mod p ),
.
. T 8 1' -1 0'1 ± 81"
.A 1 0'v - 1 - .A20'v -2 + . . . T .Av- 1 0'1 ± .Av
(mod p).
(83)
Since p is a prime greater than n , then (v, p) = 1 and the congruence (83) is soluble for 0'1" From (83) we get successively (mod p)
0' 1 == J.t1 " " , O' n == J.tn
(0 � J.t v � p - l ) ,
where the values J.t1 , . . . , J.tn are determined uniquely by setting quantities .A I , . . . , .A n . But then every solution of the system (81) coincides with one of the permutations of the roots of the congruence n X n - J.l.1X -1 +
•
.
.
± J.l. n == 0 (mod p)
with fixed coefficients and, therefore,
T(.Al ' " . , .An ) � n ! .
Now, since Tn =
p L
T( Y1 + . . . + Yn , . . · , y f + . . · + y� ) ,
yl . . . . . y n = l
we get the lemma assertion:
Tn �
p
L
yl . . . . . yn = l
n! = n! p n .
[ Ch. I, § 5
Complete exponential sums
32
Note. From this lemma and the equality (80) it follows immediately, that under any � 1 and a prime p > n the following estimate holds:
n
p I: al t···,an =1
THEOREM 7 . Let al X +
.
.
.
+ an x n .
� 2, p be a prime greater than Then
n
=
1 and f( x )
=
'
f{ x ) 2 I p- ::;; p1 n e "" I: P
(a l , " " an , p)
n,
n
-
.
x=1
Proof . At first we shall consider the case (an ' p) = 1 . Let integers >. and ft vary in the bounds 1 ::;; >. ::;; p 1 , 1 ::;; ft ::;; p. Arrange the polynomial f (>.x + ft ) in the -
ascending order of powers of x
(84) and observe that and Denote the number of solutions of the system (85) by H ( b 1 , , bn ) . It is plain, that H(b1 , , b n ) does not exceed the number of solutions of the system made up of the last two congruences of the system (85): •
•
•
•
and, therefore, since (nan , p)
=
.
•
1 and (>' , p) = 1 ,
(86) B y (25) for complete sums the equality
p 2 ,.., f{ x) ""' L.J e p •
x=1
2n =
P 2 ""' L.J e x=1
'
1ft
n f{>'X+/l ) 2 P
Ch . I, § 5]
Mordell's method
33
>. and j.L, we have p - 1 p '"' p 1. /( '\ x+ p ) 2n p 2 1rl. /( x) 2 n L: L: 2 1 p > L...J e 1r p I p p 1 ) e - '\=1 p=1 x=1 x =1 p . b 1( '\,p)x+ ... +b n ().,I')x n 2n p-1 P 1 p '"' L...J e21r 1 pep - I ) {; � x=1 G rouping the summands with fixed values b 1 ( >., j.L), , bn(>.. , j.L ) and using the esti
holds. H ence by (84) after the summation with respect to =
=
.
•
.
mate (86), we get
=
Hence by the note of Lemma 6 we obtain the theorem assertion for the case 1:
(an, p)
p ./( x ) 2 n n n ! 2n n 2 n 2 n -2 '"' 21r' L...J e P � p ep - 1 ) p < p , x=1 P 1 ' / ( x) L: e 2 1r ' -p- < np1 - n . x=1 Now we show that the general case ( a 1 , . , a n , p ) = 1 can be reduced to the case when leading coefficient of the polynomial is prime to p. Indeed, let (a8 , p) 1 and a 8 +1 == . . . == an == 0 ( mod p) , 1 � � Then we obtain P 1r ' !( x l '"' � alx + ...P+ a.x· L...J e 2 1 P L...J e21ri x=1 x=1
=
.
.
s
n.
=
The theorem is proved in full.
=
Note. A substantial improvement of Mordell's estimate was obtained by A . Weil [48] , who showed, that under prime p > n and (an , . . . , an , p) 1 the estimate
� 21ri atx + ... +a n x n L...J e
x=1 is valid.
P
� (n - I ) vp
§
[ Ch. I, § 6
Complete exponential sums
34
6 . Syste ms of congruences
One of the main points of Mordell ' s method (§ 5) is the use of the estimation for the number of solutions of the congruence system
�l . � : � '.� � � .� �� � . .. : �.�n. x f + . + x� yf + . + y� .
.
==
.
.
}
(mod p ) ,
.
where p is a prime greater than n. Hereafter, congruences of the same form but with respect to distinct moduli being equal to growing powers of a prime p will be of great importance. For the first time such systems of congruences were applied by Yu. V. Linnik [34] for the estimation of Weyl's sums by Vinogradov 's method. LEMMA 7. Let
n
� 1 , k � n( n4+1 ) , p be a prime greater than n, and let Tk (pn ) be
the n um ber of solutions of the system of congruences
l
. Yk � .� : '. � � � .� �� �.' ' : � . . . �����).n x f + . . + xi: yf + . + Yk (mod p ) ' .
.
Then
.
==
.
.
(87)
},
l=
Proof. Under n 1 the lemma assertion is evident, so it suffices to consider the case n in Lemma 5. n � 2. Take q = p, . . . , q n = p
Then we obtain
We split up the domain of the summation over
where the summation in
al, . . . , an into two parts:
L: l is extended over n-tuples al . .
.
a n which sa�isfy
and for L: 2 only those n-tuples are taken into account , for which at least for one of l v in the interval 2 � v � n p ,, - is not a divisor of a" .
Systems of congruences
Ch. /, § 6 ]
In the first case, determining b1 ,
•
•
•
35
, bn with the help of the equalities
p n 2 ' ( atz + . .. + a n z n ) p n 2 1' b t z + . . . +b n z n l: e p pn = � p L... e 11'
we get
11'1
z=1
z=1 Therefore
p n e2 11" ' ( aptz + .. . + a n ", n ) pn � L... ",=1
2k
= p2nk-2k
and using the note of Lemma 6, we obtain
an zn ) 2k p n e 2 11' 1 ( � p + + pn L ",=1 n p 2 11' i bt ",+ . ..p+ b n ", 2nk-2n l: � e �p L... bt ... . . b n =l z=l •
...
In the second case, there exists an integer
pn - l � a v p n - v . Therefore,
( a l P n -1 , a 2 Pn-2 , where 2 �
a
•
•
.
v
2n
� n! p2nk .
in the interval 2 �
, a n , pn ) = p n-I> ,
� n. But then
and using Lemma 4, we get
� e 2 11' i b p: + ...p O+ b n ", n
= L...
z=l
= pn - I>
� e 2 11'i btz + ..p.O+b n z L...
",
=1
n
v
(88)
� n such that
36
Complete exponential sums
Hence, since k � n ( n/l ) , it follows that p " 2 11"'. � e L..J
( �p +
.•.
+ a " x" p"
( Ch. I, § 6
) 2k
x=l
(89 ) Now, observing that under n � 2 n 2 k _ ( n _ 1 ) 2 k � 2k ( n - 1 ) 2 k - l
> n ( n - l ) n-l
� n!
from ( 88 ) and ( 8 9 ) we obtain the lemma assertion:
l) l) 2 n k - n ( n+ 2 n k - n ( n+ n 2 2 � !p + (n _ l ) 2 k p n(n +l) 2 n k - -2 � n 2 kp
Note. Let Tk ( P ) denote the number of solutions of the system ( 87 ) , when the domain
of variables variation has the form 1 �
Xj
� P,
1 �
�P
Yj
(j = 1 , 2, . . . , k ) .
If m is a positive integer, then under P = mpn
Indeed, using the complete sums property ( 26 ) , we get
(
mp �" 2 11"'. � + + a " x" p p" e L..J x=l •••
)
p 2 11" ' = m l: e
Tk ( mp n ) = p
n(n+ l ) p -l: 2
= m2 kp-
p mp . . . l: l: e 2 11" ' "
2
"
x=l p"
l: . . . l:
l) p n (n+-
)
(
) 2k
,
x=l
and, therefore, -
(
. a1 x + + a n x " p ... p n
"
(90)
. �+ + a n "x " p . .. p
Systems of congruences
Ch. I. § 6 )
37
Hence, since by Lemma 7
we obtain the estimate (90). run Let E�:, ... , x n denote the sum, in which the summation variables through complete residue sets modulo and belong to different classes modulo
Xl , , Xn . p L EMM A 8 . Let p be a prime greater than n , � 2, and I ( x ) = al x + . . . + a nx n . Let SOl ( al, . . . , a n ) be defined with the help of the equality . !( X l) + ... +J( x n ) 71' 1 '" 2 SOI( al , . . . , an) = pOi
•
•
•
Q
p
O
L...J
po
e
Then S01
( a1 , . . , an ) -- { p(OI - l0) n Sl ( bl , . . . , bn ) ·
a" = pOl- l b,,
if otherwise.
(v = 1 , 2 , .
. . , n),
Proof . Let us change the variables (v = 1 , 2, . . . , n ) .
Xl , . . . , Xn
Since by the assumption the quantities belong to different Classes, then the belong to different classes modulo p as well. Therefore, using quantities that
Yl , . . . , Yn
we obtain
Zl' . " , %n= l
Yl, . ·, Yn
(9 1 ) Since f'(y ) = a l +2 a 2 Y +" .+n a n yn - l , then under prime p > n and ( a l, . . . , a n ,p) = 1 the congruence f' ( y ) 0 (mod p ) can be satisfied by at most - 1 values of y from different classes modulo p. In the sum (91) the quantities Yl, . . . , Y n belong to different classes and, therefore, if ( a l, . . . ,a n ,p) = p, if ( a l , . . . , a n , p) = 1 . Y l ' ' ' ' , Yn
==
n
Complete exponential sums
38
But then by (91 ) the sum then
[ Ch. I, § 6
So-(at. . . . ,an ) vanishes under (at. . . . ,an ,p) =
(at. . . . ,an , p) = P,
1-.
If
yt .. · · ' yn
Thus
l ) if (al , . . . ,an ,p) = P t n p{ p , . ,a (a S . . lP _l n o(92) S ( al, · · . , an ) - 0 ·f ( a I , . . . , a n , P ) 1 Applying the equality (92) to So- - 1 (a lP - I , . . . , a nP - l ) we get (al , . . . ,an ,p2) = p2, S (at. . . . , an ) - { P2ns00- -2(a lP-2, . . . , anP-2) ifotherwise. _
0-
=
1
.
0-
Continue this process. Then after
a-I
step we obtain the lemma assertion:
So- (al, . . . , a n ) · f ( at , . . . ,an ,p0--1 ) = p0--1 , = { p( o- - � n 51 ( alP- ( o- - I ) , . . . , anP- ( o- - I») otherwise . . LEMMA 9 . ( Linnik's lemma ) . Let AI, . . . , A n be fixed integers, p a prime greater than and let T*(Al' . . . ' A n ) be the number of solutions of the system of congruences � 1 . � � •. • . � �� .� �� . ( �o d �) . .. n xf x: A n (mod p ) where the variables run through complete residue sets modulo p n and belong to different classes modulo p. Then n n- l ) -T*(A l , . . . , A n ) :::; n!p ( 2 n and according to the notation x a Proof. Let f (x) = alpn - l x a2p n -2x2 n of Lemma 8 1
n
},
+...+
==
+...+
+
%1 ,
Using Lemma 2 we obtain
p n
.. · , X n
n
T* ( A l , . . . , A n ) =
L II %1,
... , X n
v= l
Spv(x� . . . +
+ x� -
All),
Ch. I, § 6 )
Systems of congruences
39
where the summation with respect to al , " " a n is extended over the domain 1 � al � p, . . , 1 � an � pn . Hence observing that x�+ " ,+x� p I' 211"1. a t X l +"'+X" + .. . + a " P '"'" e pI' � X l ,···,Xn 2 11" i f(X l )+ . . . + f(xn ) pn n = 2:: e p = Sn (a l pn-l , . . . , a n ) , X l " ",Xn we have
)
(
.
T* ( A l " . . . , A n ) , n(n+ l ) p pn i a l Al + ... + a n An -211" - --'"'" S ( a n- l , . , a ) e p pn 2 =p .. n L � n lP a l=l a n=l Determine the quantities b l , , b n with the help of the equalities a l = b 1 , a 2 = pb2 , · · · , a n = p n- l b n '
(
. .
•
According to Lemma 8
Sn (a l pn - l , . . . , a n ) n(n 1 ) Sl ( b l , . . . , bn ) = p
{
�
if a" = p,,-l b ,, ( v = 1 , 2, . . . , n ) , otherwise,
where X l " " , Xn
Therefore, n(n-3) p - -2- T* ( Al , " " A n ) p L S1 (b1 , . . , ... bt ,bn = 1 P
L
P
L
e
.
2 . , bn ) e- 11" 1
b I At + ... +bn An p
. . . ·+bn (x�+ . . . + x� -An ) 2 11" i b l (XI + . . +xn -Ad+ P
Now, using Lemma 2, we obtain
n(n- l ) 2T*(A l , . . . , A n ) = p-
L II 8p ( xr + . . . + x� - A ,, ) P
Xl " " , Xn n(n-l ) p �p 2 L
=p
n
v= l
n
II 8p(xr +
n(n-l) 2 T(A l , . . . , A n ) ,
.
. . + x � - A ,, ) ,
)
.
where
[Ch. I, § 6
Complete exponential sums
40
T(Al , ' . . , A n) is the number of solutions of the system of congruences 1 ::;; X II ::;; p.
Hence, because
T(Al " . , A n ) ::;; n!
by (82), the lemma assertion follows:
.
n ( n- l )
T* (A l " . . , A n ) ::;; p-2- T(A l , ' . . , A n ) ::;; n! pn ( n- l) / 2 . COROLLARY . Let T� (mp n ) be the number of solutions of the system of congruences xj , Yj ::;; mpn , i i- j => X i ¢. Xj , Yi ¢. Yj (mod p) . 1 ::;;
Then (93)
Proof. Since each variable among X l , , Y n runs through a complete residue sys tem modulo p n m times (under the additional conditions i i- j => Xi ¢. x j , Y i ¢. Yj ( mod p )) , then using the lemma we obtain •
= m 2n
.
•
pn
L
Yt ,· . .
,Y n
2
::;; m 2np n nI p
T* ( Yl + · · · + Y n , · · · , Yr + · · · + Y� )
n(n - l ) 2
--
= n! m 2n p
2 n +l) 2n - (n 2
--
§ 7. S ums wit h exp onential function
Let form
a be an integer, m
� 2 and
q
� 2 be coprime positive integers. Sums of the P
S( P)
=
a q'"
L e 2 7U m
%=1
.
are called rational exponential sums containing an exponential function. In the inves tigation of such sums we shall need some properties of the order of q for modulus, m.
Sums with exponential function
Ch. I, § 7)
41
Let P be a prime, m = pm1 ' T and T1 be the orders of q for moduli m and m1 , res pectively. We shall show, that if T of. T1 and p\m1 , then the equality (94) hol ds. In deed, since m1 \m, then from the congruence qT == 1 ( mod m) we get qT == 1 ( mod m1 ) and, therefore, T1 \T. On the other hand, from the congruence qTl == 1 ( mod m1 ) we obtain qTl = 1 + U 1 mt , where U1 is an integer and m1 is a multiple of P by the assumption. But then qP T!
( 1 + u 1 m dP == 1 ( mod m)
=
and T\ PT1 ' Since T1 \T, T1 of. T and P is a prime, the equal h y (94) follows:
=
1 TT1- 1 \ P, TT1- = p, T PT1 . Let now m be odd, m pr 1 p� . be the prime factorization of m, T and T1 be the orders of q for moduli m and PI . . . Ps , respectively. We determine the quantities f31 , . . . , f3s with the help of the conditions q Tl 1 - U 0 pf31 1 . . pf3• (95) ( u o , PI . . . P s ) 1 . s
=
_
•
•
•
-
=
.
For definiteness we suppose that in the prime factorization of m those primes, which satisfy the inequality av > f3v , are put at the first r places (0 � r � s ) , so av > f3v under v � r and av � f3v ur�der v > r. Further let m 1 = p� l
. . . p�r p�.;:.r . . . p� • .
From the definition of T1 and the equality (95) it follows that the order of q for Ip.odulus mi is equal to 7"1 and Let us show the validity of the equalities qT
=
1 + u m,
(u ,
PI . . . Pr)
=
1 and 7"
m m1
1
= -T
(96)
.
Indeed, let m2 = pm1 ' where P is any number among the primes PI , . . . , Pr ' Let '7"2 denote the order of q for modulus m2 . Obviously 7"2 of. 7"1 ( for otherwise we would have m2 \qTl - 1 and pm1 \U1ml , which contradicts the condition (U1 , PI . . . Pr) = 1). Since, besides that, p\ml , then by (94) 7"2 = P7"l . But then qPTl q T2
where U2
==
=
=
U 1 ( mod Pl
( 1 + U 1 m1 )P 1 + U2m2 ,
==
1 + U1m2
( mod p1 " P m2 ) ,
. . . Pr), and, therefore, ( U2 ' Pl " . Pr)
'
r
= 1.
Thus
Repeating this process av - f3v times with P being equal to each Pv ( v we obtain the equality (96) .
=
1 , 2, . . , r ) , .
Complete exponential sums
42
[ Ch. I, § 7
THEOREM 8 . Let m � 2, p be a prime, m = pm 1 , T and 71 be the orders of q for moduli m and m 1 , respectively. If 7 '" 71 and p2 \m, then under any a not divisible by p Z
aq m - 0. 2: e211'i x=l T
(9 7 )
Proof . Let T denote the number of solutions of the congruence q
X
== q
( mod m ) ,
Y
1 �
X, Y
�
T.
U sing Lemma 2, we obtain
a q_'"_-=-qU-,-) e 211' i _(.:..:: m q'" 2
m a 1 2: � e 211' i m L... .J m a=l x=l T
=
On the other hand, obviously, T = 7 . But then
m aq'" 2 211' i 2: e m 2: a=l x=l T
=
mT =
(9 8 )
mT.
Therefore, b y (94)
a q '" 2 211'i m e L....J ( aa=l , p)=l x=l m = 2: a=l m
Hence for any
a
T
�
m � 211'i a q"' 2: L... .J e m a=l x=l ( a ,p) = p T
2
not divisible by p we obtain the theorem assertion T a q'" 2: e 211't. m -- 0 . x=l
Let us show that if at least one of the theorem conditions
Sums with exponential (unction
Ch. I. § 7]
43
is not satisfied, then the sum (97) might be not equal to zero. Indeed, let p > 2 be an arbitrary prime, m = 2p, ( a, m ) = 1 , and root of 2p. Then we have
11' . a q" '"'" m .L..t e 2 1 x=1
q
be a primitive
a ( 2x -1) p 211'1. ---2p = -e 211' i �2 1 . L e = x=1 x� P+2 1 In this example the condition T i- T1 is obviously satisfied, but p2 � 2p , and the second r
=
condition is violated. Let now m = p2 , 9 be a primitive root of p2 , and using the equality (98), we obtain
q
= fl . Then
T
=
P
-
1
and,
p 2 p - 1 211'i agP"2 2 a q" 2 11'i 2 m 1 L Le p >.: max L e p(p - 1 ) a=1 ( a, p )=1 x=1 ( a,p)=1 x=1 a 1g " 2 p 2 p - 1 2 11'i agP"2 2 P P -1 211'1. --1 1 '"'" L Le p -:---. � L e p = p - 1 , p( p - 1 ) a=1 x=1 p( p - 1 ) a 1 x=1 211' i a q" max L e m � JP::J. . ( a, p )=1 x=1 r
?"
=
r
In this case the condition p2 \m was fulfilled, but T = T1 . ; Another form of conditions, under which the complete sums S( T ) vanish, is shown in the following theorem. THEOREM 9 . Let m = pf1 . . . p�. be prime factorization of odd m ,
T be the order of for modulus m and the quantities P1 , . . . , Ps be determined by the equality (95). If 0 there exists v such that av > PI' and a t= ( mod p�. -p. ) , then
q
a q" '"'" e211'i m .L..t x=1 r
Proof. Chose that value
0
= .
v , which satisfies the conditions
and write a in the form a = pOl. -p. - 'Y a 1 , where 'Y � 1 and ( a ' , PI' ) = 1 . Let m = p�. - P. -'Y m', m' Pv m " , T ' and T " be the orders of q for moduli m' and m il , respectively. Since p�' \m, then pe · +'Y\m' . But then pe ' \m " and by (96 )
=
T
,
=
m' m' " - T1 = -il T " = P v T . m1 m
Complete exponential sums
44
From the divisibility of m' by pe p + 'Y it follows also, that m' = P v m " ,
r
'
Therefore, by Theorem 8
Since q r
f.
r
"
[ Ch. I, § 7
p� \m' . Thus
p! \m' and (a' , pv )
,
=
1.
a ' q�
r'
m' = 0 . l: e211'i -
x=1
==
1 (mod m) and m'\m, then q r
1 (mod m') and
==
Now, using the property (26), we obtain the theorem assertion: r
a q�
� 211'i m
L..J e x=1
T
=
� L..J
x=1
a' q �
e 11'1 -, = 2
•
T
a ' q :D
211'1 L..J e m , x=1
r'
r
m
�
'
.
=
r
'
is a divisor of
r.
O.
Note that the Theorem 9 requirements can be relaxed, namely, the condition of m being odd may be omitted. In order to prove that it suffices (see [32] ) in determining the quantities PI , . . . , Ps to use the equality q (I'H) r1
_
1
l3t fJ. - u0 p 1 . . . p s ,
where J-I. = 1 , if m == 0 (mod 2), instead of the equality (95) .
T} ==
( uo , P I " . P s) = 1 ,
1 (mod 2),
q ==
3 (mod 4), and J-I. = 0 otherwise,
THEOREM 10 . Let m � 2 be an arbitrary integer, (a, m ) the order of q for modulus m . Then the estimate
( a q., )
bX 2 11'i � m +r L..J e
x=1
=
1 , (q, m)
= i,' and
� rm
r
be
(99)
holds under any integer b. Proof. Since the fractional parts
{ �X }
and
c:}
,
have the same period T , then by (28) the sum (99) is a complete exponential sum. But then under any integer z r
2 L..J e 11" x=1
�
'
( amq., + b X ) - r
=
=
r
",,
L..J e x=1 r
2
11"
'
q"m+' + b X+bZ (a) r , qm., q. + -brX ) . (a--
"" e 2 11'1 L..J x=1
-
Ch. I, § 8]
Distribution of digits in complete period of periodic fractions
Therefore,
45
q' b�): 2 T e2 11"1. (aq'"--+....: ""T 2 11"i (aq'"-+-.!b ) 2 = LT "" L...J z=1 z=1 z=1 T e211"1. (aq"'Z bz ) 2 "" --+"" � L...J L...J L...J e
r
m
T
T
m
m
m
T
%=1 z=1
Hen ce the theorem assertion follows, because the congruence x, y T, q Z == q Y ( mod m ,
)
1� � is sat isfied for x = y only: T 2 1r1 (aq'"m + rb"') 2 T 2 1 (aq'" Z + bZ) 2 L e 11" -;n- r e � L L ",=1 :=1 z= 1 T . T "" e2 11"1. -qY)z = -rI """" L...Jy=1 e2 11"1 -L...J :=1 b ( z - y) T Om (q'" - ) m, = -:;: """" L...Jy= 1 e2 11"1 -(m ag'" + ) 2 "" 1 211" r � ..;m. e L...J z=1 1
•
m
•
T
b(z-y)
m
T
§
T
( q'"
m
m
•
qy
=
b"'
•
8. Distribution of digits in complete period of periodic fractions
Let ! be an irreducible fraction and q � 2 an arbitrary integer prime to m . In writing the q-adic expansion of the number ! , the following infinite pure recurring "decimal" to the base q arises:
: [ : ] 0 ·")'1 1'2 . . ")' +
=
.
z · . .
")'''' + T = ")'z
,
(
x
�
1 ),
( 1 00)
with a period T being equal t o the order, t o which q belongs for modulus m . Let N!:) ( 01 . . denote the number of the times that the following equation is satisfied: .
On )
")'z +n = 01 . . . On ( x 0, 1 , . , P - 1 ), . . . On i s an arbitrary fixed n-digited number i n the scale of In On) is the number of occurrences of the given block 01 . . . On among the first P blocks , ")'P+n-1 , ")'1 ")'n , ")'2 ")'n+1
"),,,, + 1 where P � and 0 1 other words, N!:) (01 . r
of digits of length
n
•
.
.
=
•
.
q.
. .
•
•
•
•
•
•
,
•
.
•
,,),p
•
•
•
[ Ch. I, § 8
Complete exponential sums
46
formed by successive digits of the expansion ( 100). The question about the nature of the distribution of digits in the period of the fraction 1il is closely connected with properties of rational exponential sums contain ing exponential function. This connection is based on the possibility to represent the quantity N!'; ) ( 81 8n ) in terms of the number of solutions of the congruence •
•
•
a q X ==
y
+ b ( mod m ) ,
o
�
< P,
x
1�
where b and h depend on a choice of the block of digits 8 1 number of solutions of the congruence ( 101) by T!'; ) ( b, h) .
y •
.
� h, •
( 10 1 )
8 n . We denote the
LEMMA 1 0 . Let quantities t , b , and h be defined by the equalities b=
[:�],
Then Proof. Let x be any solution of the equation
(0 � x < P ) .
( 102)
Then we obtain from (100)
{ } aq X m
Ox = O.1'x+l . . · 1'x + n . . . = O .1'x + l . . · 1'x + n + -n q t + Ox Ox = -= O. 81 . . . 8n + n , qn q
where 0 < Ox < 1. Hence it is plain that the equality ( 102) is satisfied for those and only those x , for which O � x < P.
( 1 03)
Since from the definition of b and h it follows that
� m
�
� qn
r
(0 �
r
� s ) . Choose
O' r + 1 A. m 1 - P1Ih " ' PrPr Pr+l · · · Ps . _
Then the order of q for modulus m 1 should be equal to T1 and T = :::1 T1 by (96) . LEMMA 1 1 . Let b == bI (mod mt ), h == h 1 ( m od m 1 ) ' and h � h 1 . Then
Proof. Using Lemma 2 we get T!,;) (b, h ) =
T- l
h
L L 6m (aqX X= o y=1 m ""'h 1 ""'
= - L...J
m z=1
(
L...J e
y=1
-
.
y - b)
(YH)Z ) (
- 2 71'1 -
m
T
""'
L...J e
a zq
",
2 71' -
m
)
•
x=1
By Theorem 9 the inner sum of the right-hand side of this equality m ay not vanish only for values z, which satisfy the congruences
v = 1 , 2, i.e. , for
.
.
.
, r,
[ Ch. I, § 8
Complete exponential sums
48
Therefore, using r
=
..!!L ml r1 and
b b1 ==
( mod m 1 ) , we obtain
x=1 y=1
x=1 y =1
Since the difference h - h l is a multiple of m 1 and, therefore,
h h-h l h - h1 X b ( b = q , ) a - Y 1 L bm1 ( Y ) = -L m1 m1 y=1 y =ht + 1 -
we obtain the Lemma assertion: Tt hl
T� ) ( h ) =
L L bm 1 ( a q X - Y
-
Tl bd + L x=1 y=h 1 +l
Let us consider the question concerning the distribution of blocks of digits in the complete period of the fraction ;; . Since there exist q n distinct blocks b 1 . . . bn, then the mean value of the number of occurrences of a given block of n digits equals r.
in
THEOREM 1 1 . Let value:
Rn be the deviation of the quantity N�) ( b1 . . . bn ) from its mean R N� ) ( b1 . . . bn ) = �T q n + n.
Then under odd m , any n � 1 , and any choice of the digits estimate is valid:
b1 . . . b n
the following
(106) where rl is the order of q for modulus being equal to the product of primes entering into the prime factorization of m . Proof. By Lemma 10
where
N!:) ( b1 . . . bn ) = T�)(b, h ) ,
T�) (b, h ) is the number of solutions of the congruence o � < T, 1 � Y � h , a q X == Y + b ( mod m ) , x
Ch. I, § 8)
Distribution of digits in complete perio d of periodic fractions
b and h are determined with the help of the equalities
an d the quantities
[:7],
b= Let
49
h1 denote the least non-negative residue of h to modulus m1 : h
Observing that
==
h1
0 ::;; h < m1 .
( mod mI ) ,
h��' is an integer and
h we obtain
=
m
qn
_
{(t +qn1 )m} + { tmqn },
[ ] [
h - h1 m m h - h1 = q q n n � m1 - m1 �
I
]
But then it follows from Lemma 1 1 by virtue of the equality
Since 0
::;;
h1
::;;
ml
m1 - 1 , then
if h1 = 0 and otherwise and therefore
7 = .!!!:.. 71 , that
tm } { (t +qri1 )m } { (j" ,
T�) (b , h) - ;n 7 = T�:) (b1 ' h I ) - { rn%n } 71 +
{
71 0
if h1 0 and otherwise.
=
' 1 ,
n,
.
.
.
, >'r are distinct roots of the characteristic equation ( 1 09)
and PI (a;), . , P r ( a; ) are polynomials whose degrees are unity less than the multiplic ity of the corresponding roots of the equation (109). In particular, if the characteristic equation has no multiple roots, then .
.
( 1 10) where C1 , , en are constants depending upon the choice of initial values of the function "p ( a; ) . If coefficients of the equation (108) and initial values "p ( 1 ) , . , ,,pen) •
•
•
.
.
Complete exponential sums
54
[ Ch. I, § 9
are integers, then, obviously, under any positive integer x the function 1jJ( x ) takes on integer values. Let m > 1 , ( a n , m) 1 , and at least one of the initial values 1jJ(1), . . . , 1jJ(n) be not a multiple of m. In the equation (108) we replace x by x + n and transit to the congruence to the modulus m :
=
1jJ (x + n) == a 1 1jJ (x + n - 1) + . . . + a n 1jJ (x) (mod m).
( 1 11)
Since ( a n , m ) 1 , so in this congruence 1jJ(x) can be expressed in terms of 1jJ(x + 1), . . . , 1jJ( x + n ) and, setting x = 0, - 1 , -2, . . . , we may extend the function 1jJ( x) for integers x � O. A function 1jJ(x) determined for integers x by the congruence ( 1 1 1 ) and initial values 1jJ ( 1 ) , . . . , 1jJ(n) (see [21] ) is called a recurrent function of the n-th order to the modulus m, and the sum p , ,,, ( x) S (P ) = :E e 2 1f1 rn x=l
=
a recurrent function. It is easily seen that under n = 1 these sums coincide with considered in § 7 sums with an exponential function. Let us show that a sequence of least non-negative residues of the function 1jJ(x) to modulus m is periodic and that its least period does not exceed the quantity· m n - 1 . In fact, let us denote the least non-negative residue of 1jJ(x) to modulus m by 'Yx :
. an exp onential sum with
.,p(x) == 'Yx
(mod m ),
o � 'Yx � m - 1 .
Then by virtue of ( 1 1 1 ) 'Yx + n == a l'Yx + n
-
+ . . . + a n'Yx
l
(mod m).
( 1 12)
Consider blocks of n digits with respect to the base m (x = 0, 1 , . . , m n ) .
'Yx + 1 . . . 'Yx + n
.
( 1 13)
Since the number of distinct blocks of n digits is equal to m n , then among the blocks ( 1 1 3) there exist two identical blocks ( 1 14) We determine
r
by the equality
r
=
X2
'Yx + 1 . . . 'Yx + n
- X l and will show that under any x �
= 'Yx + r+ l . . . 'Yx + r + n·
Xl
( 1 15)
Exponential sums with recurrent (unction
Ch. I, § 9 ]
55
In fact, under x = Xl this equality is fulfilled by virtue of ( 1 14) . Apply the induction. Let us suppose that the equality (1 15) holds for a certain x � X l . In the congruence ( 1 12 ) we replace x by x + 7 + 1 . Then using the induction hypothesis we obtain
')'z+r+n+l and, therefore,
a l')'z+r+n + . . . + a n')'z+r+l a1"Yz+n + . . . + a n')'z+l == ')'z+n+l
== =
')'z+r+ n + l
=
')'z+n+ l .
(mod m ) ,
But then
')'z+2 . . . ')'z+n+ l
= ')'z+r+2
. . . ')'z+r+n+ l ,
hence the equality (1 15) is proved for any x � X l . By me ans of such considerations we get this equality for x < X l as well (but in this case, ')'x should be expressed from t he congruence (1 12) in terms of ')'x +l , ' . . , ')'z+n beforehand, and that could be done because of ( a n , m ) = 1 ) . Hence it follows that the least non-negative residues of the function 1/J( x ) have a period 7 , where 1 � 7 � m n . Let us assume that the least period is equal to m n . Then any block of n digits should occur among the blocks (1 13), and, in particular, the block 0 . . . 0 being formed by zeros only is present among them. But then by ( 1 12) all terms of the sequence of the residues will equal zero and its least period equal that contradicts to the assumption. Therefore, the least period of the function 1/J( x ) does not exceed m n - 1 . Henceforward we let 7 denote the least period of the sequence of the least non negative residues of the function 1/J( x ) to modulus m. It is easily seen that 7 is the period of fractional parts of the .function "'�) :
1,
Therefore, the sum
S(7)
=
r 2 . ---;n x) :�::> 71"1 ",( x =l
is a complete exponential sum. Since under integer
then by (28) under any integer
a the sum r ' ("'(X ) + B X ) ""' 2 L.J e
z=l is a complete sum as well.
71"1
m
r
a
[ Ch. I, §
Complete exponential sums
56
9
Let .,pI ( X ) , . . . , .,p n (x) be recurrent functions satisfying the equation ( 108) and de termined by initial values if x = j, if 1 � x � n,
x i= j
(j = 1, 2, . . . , n ) .
It is easy to show that .,p(x + z) = .,p(z + l ).,p l ( X ) + . . . + .,p(z + n).,p n (x).
( 1 16)
In fact by virtue of the linearity of the equation ( 108) , any linear combination of its solutions is a solution too . In particular, the sum in the right-hand side of the equality ( 1 16) is a solution of the equation (108). From the definition of the functions .,pj (x) it is seen that under x = 1 , 2, . . . , n the initial values of this sum are equal to .,p(z + 1) , .,p ( z + 2), . . . , .,p (z + n ) , respectively. The solution .,p(x + z) has the same initial values. But solutions, which have the same initial values, coincide. Hence the equality ( 1 16) is proved.
THEOREM 1 3 . Let .,p(x) be a recurrent function of the n-th order to the modulus m,
'T
be its lea.s t period, and P � T
•
� 2 11"1
L.-J e
x=1
x
",( ) m
T.
Then we have the estimates P
n
Le x=1
� m2 ,
Proof. Since under an integer
a
",(x) 2 1I"1. --;n
!!.
� m 2 ( 1 + n log m).
the sum
is a complete sum, then under any integer z
( ",( x +z) + a x +z ) T m , x=1 ( "'(x + z) + "T ax ) mISa(T) 1 L /1r1 x=1 Sa(T )
T
=
L
e
11" 2 i
T
•
=
Squaring and summing over z yields T-l
T ISa( TW
=
L
T
Le %=0 x=1
. ( ",( x +%) + a x ) 2 11"1 m
-- "T
2
I
( 1 1 7)
Ch. /, § 9]
Exponential sums with recurrent function
57
We let 'Yz denote the least non-negative residue of the function ,p(z) to modulus Then by ( 1 16 )
m.
,p(z + 1 ),p1 (X) + . . . + ,p(z + n),pn(x) == 'Yz+ 1 ,p1 (X) + . . . + 'Yz+n,pn(x) (mod m),
,p(x + z)
=
and, therefore,
T -n '" 2 11'1. 1/1(X) L...J e -m = I Sa( T ) 1 � m 2 • ( 1 18 ) x=l Since T is the least period of 'Yz to modulus m, then under z = 0, 1 , . . . , T - 1 all block s 'Yz+ 1 . . . 'Yz+ n of n digits are distinct . Therefore, exlending the summation to all p ossible blocks Zl . . . Z n of n digits, we obtain
p-1 T I Sa ( T ) 1 2 � %1 ,
L •••
, %n = O
T
=
a (x-y)
2 71'1 -'" T L...J e •
x ,y=l
� mn
T
m -l L
%l , . . . , Zn =O
L 15m [,pI (x) - ,p1 (Y)]
x , y= l
.
. .
15m [,pn (x) - ,pn (Y)]
=
m n T,
( 119 )
where T is the number of solutions of the system of congruences
: ¢1 (�� }
�
�1 ( ) ,pn(x)
=
,pn {Y)
(mod m ) ,
1 � X , Y � T.
( 1 20 )
Let us assume that this system has a solution with Y -=I x . Without loss of generality, we may assume that Y > x . Using the equality ( 116 ) , we get
,p{Z) Y ,p{z + - x)
,p{z - x + 1 ),p1 {X) + . . . + ,p (z - x + n),pn{x) , = ,p{z - x + 1 ),p1 {Y) + . . . + ,p(z - x + n ) ,pn (Y) .
=
Hence by (120 ) it follows that under .any integer z
,p(z + Y - x)
==
,p(z) (mod m).
But then Y - x is a period of 'Yz , and since T is the least period, then Y - x � T , which leads to a contradiction. Thus, the congruence system ( 120 ) has no other solutions except for solutions with Y = x and, therefore, T = T . Now from ( 119 ) , we get
( 121 )
Complete exponential sums
58
Hence under
a =
[ Ch. I, §
9
0 the first assertion of the theorem follows:
T 1/I(x) L: e 2 '1r1 --;n x=1 •
=
!!
ISo( r)1 � m 2
.
The second assertion of the theorem follows immediately from Theorem 2 and the estimate ( 12 1 ) :
T ' ( 1/I( X ) ax) P . 1/I(x) L: e2 '1r 1 --;n � max L: e 2 'lr l --;n + T (1 + log r) l �a�T x=1 x=1 n = max I Sa (r) l ( l + log r) < m 2 ( 1 + n log m) . l �a�T Note that in the general case the order of the estimation
can not be improved further. Indeed, using considerations from the theory of finite 'fields (see, for instance, [33]) , it can be shown that under any prime p > 2 and positive integer n < p there exist recurrent functions t/J(x) of the n-th order to the modulus p with the period r = p n - 1 . Besides, roots of the corresponding characteristic equation ( 1 09) are distinct and by ( 1 10) •
( 1 22) By virtue of properties of symmetric functions there exists an equation with integral coefficients J.t n b 1 J.t n- 1 + . . . + b n ,
=
whose roots J.t l , . . . , J.t n equal A � , . . . , A � , respectively, and the free term is relatively prime to p. Consider the functions t/J (2x) and t/J(2x + 1 ) . It follows from ( 122) that t/J(2x) t/J(2x + 1 )
= C1 J.t f + . . . + CnJ.t!, =
C1 A I J.t f + . . . + Cn A n J.t !.
Thus, t/J(2x) and t/J(2x + 1 ) are recurrent functions of the n-th order t o the modulus satisfying the equation
p
t/J*(x)
=
b1 t/J*(x - 1 )
+ . . . + bnt/J*(x - n).
Denote by "Ix the least non-negative residue of t/J(x) to modulus p . Under rl = ! r we get * "I2( X+ Tt } "I2 X+ T 12 x and, therefore, 12x has a period being equal to rl .
=
· Since p is o d d , then
=
T:! = pR2- t
is an integer .
Exponential sums with recurrent function
Ch. I, § 9 ]
1'2
59
Let us assume that 1'1 is not the least period. Then we can find a positive integer < 1'1 , such that under any integer x ( 1 23)
A pplying the equality ( 1 16) we obtain
+ 1) + . . . + .,p (2x + 1 ).,pn (2 2 + 1 ) , (1) + . . . + .,p (2x + n - 1 ) .,pn ( I ) .
.,p (2 x + 21'2 ) = .,p (2 X) .,pl (21'2 .,p (2x) = .,p (2x ) .,pl
n -
1'
But then by ( 123) the congruence
should be fulfilled under any integer x . From properties of solutions of the system (120) , it follows that at least one of square brackets in ( 1 24) is not congruent to zero to modulus p and, therefore, the number of solutions of the congruence ( 124) does not exceed p n - l - 1. On the other hand, according to the definition of the function ?jJ (x) under x = 1 , 2, . . . , 1'1 n-tuples 12x , 12x + l , . . . , ')'2x + n -1 yield distinct solutions of this congruence. Since obviously
1'1
=
n-1 p2
> pn-1 - 1 ,
tpen w e arrive at a contradiction and, therefore, 1'1 i s the last period of ')'2 x Analo gously we get that the least period of ')'2x + 1 is equal to 1'1 as well. Now, in the same fashion as in the deduction of ( 1 18) , we arrive at the equalities •
Tl
•
2 '11' 1 L.J e 1'1 "" x=1
!JI(2x+1) 2 P
_
Tl -1 "" L.J z=O
Hence by virtue of the choice of the function ?jJ( x ) , it follows that
1'1
(
)
2 . 1/>(2x) 2 2 71' 1 -271'1. 1/>( 2x+1) "" "" P e e P L.J + L.J x=1 x=1 }I",, zl 1/>1 (2x)+". + zn 1/>n (2x) 2 - I' 2 '11' 1 "" = P , L.J e L.J Zl , , , . , Zn =O x=1 Tl
TI
TI
•
( 125)
Complete exponential sums
60
[ Ch. I, §
I
Z1 ,
9
, Zn, where the sign I in the sum L: z 1 " " , Z n indicates the deletion of n-tuple formed by zeros entirely, from the range of summation. Let denote the number of solutions of the system of congruences
T1
1
1
� ?�). �.� ?�)
1P n(2 x ) == 1P n(2y )
(mod p)
}
In the same way as in the system ( 120) , we have
p-1
1 � x, y �
T1
71
=
•
•
•
7.
and, therefore,
L
%1 " " , Zn =0
But then (125) can be rewritten in the form
71 TI
(
:1: = 1 '"""
L....J
We determine
TI
L....J e
1
'"""
x=
.
•
2 11'1
t/t(2x)
2 7f 1 -P e
I S* ( 7d l
--
t/t ( 2 :1:) 2 P
TI
+ TI
2
+
:1: = 1
L....J e '"""
.
7f 1 '""" e 2
:1: = 1
L....J
•
2 7f1
t/t ( 2 :1:+1 ) 2 P
t/t(2 :1: + 1 ) P
2
=
P
)
n - 71
=
-1 ' -+ 2
pn
(126)
with the help of the equality
Then from (126) we get
1 .!!:. I S*(7dl > "2 p 2 . Hence by ( 1 17) it follows that under any prime p > 2 and any n > 1 there exists a recurrent function of the n-th order to the modulus p such that for the exponential sum S *( 7t ) the following estimates
1
.!!:.
"2 p 2 < hold.
I S* ( 7d l
.!!:.
� p2
Ch . I, § 1 0 ]
Sums of Legendre's symbols
61
§ 10. Sums of Legendre's symbols Let p > 2 be a prime, I ( x ) ao + al x + . . . + a n x n be a polynomial with integral coeffi cients, n < p , and ( a n , p) = 1 . Let U n denote the sum of Legendre ' s symbols =
Un = t x=1
( I(pX) ) ,
( 127)
and Tn denote the number of solutions of the congruence
y2 The quantities
==
I ( x)
( mod p) .
( 128)
Tn and U n are connected by a simple relationship p Tn = � 1 + I ( x ) p + Un . P
[ ( )]
=
( 1 29)
This relationship reduces the question on the number of solutions of the congruence (128) to studies of sums of the Legendre symbols. The sums ( 127) are easily evaluated for polynomials of the first and the second degree. Indeed, since Legendre's symbol ( : ) is a periodic function with a period p and under (a I , p ) = 1 the linear function ao + al x runs through a complete residue system modulo p , when x runs through a complete residue system modulo p, then
t ( ao + a1 x ) t ( :. ) p P =
x=1 In order to evaluate the sum 17
we consider the congruence
2
=
o.
x=1
=
t ( ao + a X + a2 x2 ) , l
p
x=1
y2
==
x 2 + a (mod p)
and denote the number of its solutions by
p
T( a ) .
Obviously,
T ( a ) = L Dp ( X 2 - y 2 + a) x , y=1
X2_ 2 1 p - l e 2 11'1. a % P e 2 11' 1' Z( p y ) =-L P L P z= o "" y= 1 l 2 2 1 p- e 2'11'i � P e 2'11'i !!P L L p = P + pz= 1 x=1
[ Ch. I, § 1 0
Complete exponential sums
62
Using the fact that the modulus of the Gaussian sum equals -/p , we obtain
a + p-lL.,.. €2... p
T( ) = p and, therefore, by ( 1 29)
P
Let
a2
L :c = 1
. az
'"'
z= 1
(X2 +a ) -p
a
= p + pop ( ) - l ,
a
= pop ( ) - 1
( 130)
.
¢. 0 (mod p) . Then observing that
( � ) ( 4 ;2 )
= 1,
we get
( 131 ) Note that , in particular,
� (x - a ) (X - b) L.,..
:c=1
-p
p
--
= pop(a - b) - 1 .
( 132)
Indeed, this equality follows at once from ( 1 31 ):
ab - (a+ b)x + x2 ) (a [4 ab - (a +
= t( ) � ( ) � ( t p p :c= 1 :c = 1
= pOp
Under n � 3 the investigation of the sums for some special cases.
p
(T n
b) 2 ] - 1 = pOp
- b) - 1 .
is much more complicated, except
Sums of Legendre's symbols
Ch. I, § 1 0]
Consider one of such special cases. Let n � and
63
3 be odd, p
>
n be a prime,
Let us show that
( aI, p) = 1, ( 1 33 )
zx
runs through a complete In fact , since under z ¢. 0 ( mod p) the linear function residue system modulo p when runs through a complete residue system modulo p, then
x
Therefore,
l un (al Z n - 1 )1 = t ( z n x n +p a1 z n x ) t ( x n +p a1 x ) I un ( ad l . x= 1 x= 1 Squaring this equality and summing over z, we obtain p- l n 2 p - l (134) (p - 1) l u n ( a dl 2 = L l u n (a lZ - 1 ) 1 = L teA) I U n ( A ) 1 2 , %=1 �= 1 where t( A) is the number of solutions of the congruence alzn - 1 A (mod p) . Since teA) � 1 , then from (134 ) it follows that pL- l (p - 1) l u n ( a l ) 1 2 � ( n - 1 ) I U n ( AW �= 1 n + Ax y n + AY = (n - 1 ) � t ( x �=1 X, I/= 1 P ) ( P ) t ( A + xn-1 ) ( A + yn- 1 ) 1 ) x,�l/= 1 ( XpY ) �=1 =( P . P Hence, using the equality (132), we get the estimate (133 ) : p-l Y n d ( l un a l 2 � p = � X,I/L= 1 ( Xp ) [p.sp (x n - l yn - 1 ) - 1] ( n __ �P � xY .sp ( n - 1 y n - 1 ) � ( _ 1)2p, = p X, 1/= 1 ( P ) x l un ( adl � ( l )JP =
=
==
n
-
n
_
_
_
n
-
.
n
Complete exponential sums
64
Under odd
n
� 3 and
( an , p) = 1
[Ch. I, § 1 0
the same estimate holds for the general case also: ( 1 35 )
For n = 3 this estimate was obtained by Hasse [ 1 3) , under an arbitrary n it follows from more general results of A. Weil [48) . One can acquaint with elementary methods for obtaining estimates of the sums ( 1 35) by papers [35) , [42) , and [31 ) . A s i t was shown above, sums of Legendre's symbols for polynomials of the second degree can be evaluated with the help of Gaussian sums. Let us show that G aussian sums can be used in estimating the simplest incomplete sums of Legendre ' s symbols:
O'(P ) = x=L1 ( :.P ) (P p
vP log p this estimate is better than the trivial one. The availability of a nontrivial estimate
� (�) P P
' ) the number X n = >.. Then under any e (0 < e � 1 ) we have •
.
•
where the constant Cn (e) depends on n and e only. Proof. Let
a
� 1, p �
2,
and 0
< e
� 1 . Since
1 + ae log p
. � 2 be given by its prime factorization: (P I < P2 < . . . < P . ) · 1 ::;; Pr+ I ' Then, applying the Estimate the number of divisors of >.. Let Pr < estimate ( 147) for P I , . . . , Pr and using the estimate ( 148) for Pr+ 1 , . . . , P s , we obtain
ee
1
Hence, because the number of primes, which are less than e e , does not exceed follows that r( >') ::;;
C� ) l g2
1 e€
, it
1.
e
e
>. e = C (c)>.e .
Replace c by � in this estimate. Then, observing that rn ( >') ::;; [r( >. )] n , we get the assertion of the lemma:
LEMMA 14 . Let P � 2 and a () a--+ q q2 '
( a, q) = 1 ,
I()I ::;; 1 .
Then under any positive integer Q and an arbitrruy real f3 we have
1°.
t ( t, (
min p,
x-I
2° .
min p 2 ,
lI a X
� ) ( �) : ) ( �)
lIa x
f3I1.
(3 11 2
::;; 4 1 +
::;; 4 P 1 +
(P + q log P) , ( p + q).
Weyl's method
Ch. tI, § 1 1 ]
Proof.
Let us represent f3 in the form
b + (h ,
f3 = q
b
73
-q
1 (1t I
< 1 and the sign of (} 1 is opposite to the sign of (}. Then where is an integer, un der 1 :;:;; :;:;; q we obtain
x
I
ax + b + 81 8 x2 + -, a x + f3 = -q q ax + b = ax + f3 + (} q
q
I lI
-
(:� : ) I :;:;; lI ax
4-
f3 1 + � .
(149)
At first we shall show that
Indeed, if q or
�
min
(p, lIax � (3 1 )
:;:;;
3P + 4q log P.
( 1 50)
P is less than four, then this estimate is trivial. x, under which ax + b �
P � 4. Then, according to (149) for those values
I
w�
have
l I a x + f311 �
l I ax b I .
�,
q
+ q
Let q � 4 and
_
� � � I a x q+ b I .
This estimate may be used for all x within the interval 1 :;:;; of those, for which + == 0, ±1 ( mod q ) .
x
:;:;; q with the exception
ax b
Since q ) = 1 , then runs through a complete residue system modulo q, when runs through a complete residue system modulo p. Therefore
ax + b
( a,
x
t
x=1
min
(p, lI ax � ) (3 1 1
:;:;;
3P +
L
a x + b�O , ± 1
= 3P + L
2 �x '1+ ... + n >' n ) d�' 1 , there corresponds one definite aggregate of so lutions of the system ( 1 67) and each solution of the system enters into one and only one of t hese aggregates. Thus, considering all possible n-tuples >'1 , . . . , >' n , we get all solutions of the system (167) and, therefore,
. . . , >'n
L... � [N�P)(>'I , . . . , >'n)] 2 N2 k (P). =
�l ,
n
,
In investigating properties of the system of equations
��.:. : : : � .�k. �.� � } ,
x� +
.
( 1 68)
. . - Yk >' n =
. and in deducing estimates of Weyl's sums, the relationship between the exponential sums
S ( '11
kPII ,
( 1)'111
We shall show that the above established properties ( 163)-( 165) of quantities N� P) (>' 1 , . are evident corollaries of this expansion. In fact, setting 1 = . . . = = 0 in ( 169) , we obtain the equality (164):
. . , >'n)
a
an p2 k = L N� P) (>'b " " >'n). � 1 , . . . , ). n
Systems of equations
Ch. /I, § 1 2]
83
, an ) 1 2 k .
The equality (165) follows at once from Parseval's identity for the function IS(al ! ' " : 1
L: [Nt) ( A l ! " " An )f = J
'>'1,·
0
.. , '>' "
=
Finally, setting
an
=
1
.
.
.
1
f [ I S(a l! " " an ) 1 2 k f dal . . . dan 0
1
k dal . . . dan N2 k ( P) . , an)1 S(a , 4 " I l ' f···f =
o
0
0 in (169) , we get
[
� '" )1 2 k - � LJ Nk( P ) ( \ LJ IS( '-"l'" , · · · , '-"n-1 '>'1, ... ,.>.,, - 1 .>. "
]
\ ) e211'i ( O'I'>'I + "'+ O'''_I'>' ,, -d
AI , · · · , An
Hence by virtue of the uniqueness of the expansion of the function in the Fourier series
IS( al , . . . , an - l ) 1 2 k
=
� Nk( P) ( \ '>'), ... ,.>.,, - 1 LJ
.
IS(a l , . . . , an_I)12 k
\ ) e211'i ( 0' 1 '>' 1 + ... + 0',,-1.>.,,- 1 )
AI , · · · , An-1
the equality (163)
follows. The most important question in the theory of the systems of equations
��.� : : : �.�� .� . �
xf +
. . - Yi: = 0 .
},
is a question concerning the character of the growth of the nwnber of system solutions in dependence on the magnitude of an interval of the variation of variables, i.e. , a question concerning the character of the growth of the quantity while increases infinitely. It is easy to establish a lower bound for � (j = Indeed, since 1 � 1 , 2 , . , k ) , the quantities ways. Choosing then YI = , can be chosen in Yk = Xl , we obtain solutions. Therefore we have the estimate .
.
.
•
•
,
Xk
X lpk, X k •
•
•
Nk ( P) . pk
Nk (P) P Xj P ( 1 70)
Weyt's sums
84
[ Ch. II, §
12
Next, by ( 162) and (164)
p2k =
L
Nk P) P' l , " " A n ) � Nk (P )
L
>.t . . . . . >.n
1�
n(n+1) ( 2k ) n P -2- Nk (P ) ,
1>'. I< k p· and, therefore,
Nk (P ) � ( 2k1 n P )
1) 2k- n ( n+ 2
Taking into account this result and the estimate (170), we get the lower estimate for
Nk (P )
( 171) We shall show that under k � n this estimate indicates the precise order of the growth of the quantity Nk (P). Indeed, consider the system of equations
}
�� .:. . . . � �. . � .�� : :::.:. :� , '''
k
x � + . . . + x t = yf + . . . + y:
( 1 72)
In the same way as in the proof of Mordell's lemma (§ 5), it is easy to verify that the quantities satisfying this system coincide with permutations of quantities Since the system , Y k . Hence the number of its solutions does not exceed k ! ( 1 72) is obtained from the system
Yll ' . .
Xl , . . . , Xk
pk .
}
�l � : : : .: . �� �. �� . : � � � � .�k. . . . , x f + . . . + x i: = yf + . . . + Y i: by omitting the last n - k equations, then Nk (P ) does not exceed the number of solutions of the system (172) and, therefore, Nk (P) � k ! Thus, under k � n the estimate (171) has the precise order with respect to P. Under k � n it is easy to show that Nk (P ) � n! Indeed, using the trivial estimation of the sum we get
pk . p2 k - n .
S( a 1 , " " a n ) 1 Nk ( P ) = J . . . J I S( a1, ' " , a n W k da 1 . . . da n o 0 1 1 2k � p - 2 n J . . . J I S ( 0' 1 , . . . , a n W n da 1 . . . da n o 0 2 2 n k Nn (P) � n! p 2k - n . =p 1
( 1 73)
85
Systems of equations
Ch. /I, § 1 2)
Nk (P),
having under k > n a precise order with A question on upper estimates of is much more difficult. This question, which is referred to as the mean s to t ec e p r value theorem, is main in a method suggested by Vinogradov to estimate Weyl's
P,
sums .
In proving the mean value theorem we shall need two lemmas. LE MMA 1 5 . Under any fixed integer a , the number of solu tions of the system of eq u ations
��� : �� � ::: � �:� : �� � �
},
(X l + a) n + . - ( Yk + a) n = 0 does not depend on a and is equal to Nk (P) . , Y k be an arbitrary solution of the system of equations Proof. Let Xl , .
.
.
.
.
.
•
•
.
.
.
.
.
.
.
•
�.1 .� : : : �.�� .� .� } ,
xf + . .
Then under any
( 1 74)
s
=
1 , 2,
.
.
.
.
- Yk
=
( 1 75)
0
we obtain
, n
B (X j + aY ( Yj + aY = L C:as -lI (xj - yj ) , 11= 0 -
k
L ((Xi + a )S - ( Yi + a) B ] i= l
=
S
k
L C: as -II L (xj yj) 11= 0 i= l -
= 0,
where C: denotes the number of combinations of s objects v at a time. Therefore each solution of the system (175) is a solution of the system ( 1 74) . It is just as easy to verify that in its turn each solution of the system ( 1 74) is a solution of the system ( 1 75). But then these systems of equations have the same number of solutions, and this is what we had to prove. Note. According to Lemma 15
a + P e2'11'i (alx + . . . +a xn ) 2k n L l x =a +
and, therefore, under any integer 1 1
a the equality
a + P e2'11'i (a1x + .. . +an X n ) 2k . . . dX1 . . . dX n J J L o
0 x =a +1 I
=
I
" e211' i (alx+ ... +Q n x n ) 2k dXl . . . dX n p
Jo J0 x =l .
..
L...,;
Weyl's sums
86
[ Ch. II, §
12
holds. Let, as in § 6 (the note of Lemma 7) , Tk (P) be the number of solutions of the system of congruences
.�� .� .. . ... �. �� . � � . . (��� :: .
.
X l + . . . - y;
==
.
.
0 ( mod p n )
},
( 1 76)
We shall show that the number of solutions of this system can be expressed in terms of the quantity Nt) (>' 1 , . . . , A n ) ·
LEMMA 1 6 . We bave tbe equality Tk ( P ) =
L
>'1 , . . .
,An
NiP) ( A lP, . . . , A n pn ) ,
wbere tbe summation is extended over tbe region
( 1 77) Proof . It is easily seen that the congruence system ( 1 76) is equivalent to the totality of the systems of equations
�
�
�l . . . � �� � l � . . .. . . X l + . . . - y; = A npn .
.
.
.
},
arising under all possible n-tuples of integers A I " ' " A n . Since under fixed values A I , . . . , A n the number of solutions of this system is equal to Nk P) (A l P , " " A n pn ), then the sum of the quantities NiP) (AlP " . . , A n pn ) , extended over all possible values AI , . . . , A n is equal to the number of solutions of the system of congruences: Tk( P ) =
L Nt) ( Al P ' " ' ' A npn ). Al , . .. ,A n
It is sufficient to carry out the summation over the region ( 1 77) , because otherwise at least for one value v ( 1 � v � n) the inequality IA"p" l � k P" would be fulfilled and the corresponding summand NiP) ( Al P , " " A n pn ) would vanish .
Ch. 1/, § 13)
Vinogradov's mean value theorem
87
§ 13. Vinogradov's mean value theorem As
it was said in the preceding section, the mean value theorem pursues the aim est ablish an upper estimate for the quantity Nk(P), where Nk (P) is the number of in tegral solutions of the system of equations to
� 1 � ::: :. �� .�. �� . �. :'. : � .Y.k.
} , . x f + . . . + x � = yf + . . + Yk ..
.
1 � Xj , Yj � P.
( 1 78 )
The proof of the mean value theorem, suggested by I. M. Vinogradov, is based on recurrent process reducing the estimation of the quantity 'Nk(P ) to the estimation and PI < P. Two proofs of the mean value theorem are < of Nkl (P1 ) , where sented below. The first of them is simpler, but it leads to a result , which is valid repre only under an over-abundant number of variables in the system ( 1 78). The second proof is a bit more complicated, but it enables us to obtain results which are close to final ones. Both proofs are carried out with the help of different variants of the p-adic approach suggested by Yu. V. Linnik [34] for this problem. a
kl k
PI
=
estim ate
holds.
1
1
n � 2, = n2 , P > n n , p be a prime, p 7i � 2P 7i , and . l n p - Then under k > n 2 for the number of solutions of the system ( 1 78) the
L EM MA 1 7 . Let
r
P
n 2 r ,
(
)
1 n 1+n-1 P>n , r
Gr
_ -
•
( .!.)T
n(n + 1 ) 1 -n 2
•
Then for the number of solutions of the system (178) we have the estimate
( 180) Proof. The assertion of the theorem can be obtained with the help of rather simple i n duction. Indeed, under r = 0 the estimate ( 180) is trivial. Let it be true under a certain r � O. Choose
k > n2 ( r + 1 ) Determine positive integers
and
( -)
n 1 1 P > n + n-1
r+l
r, P1 , and a prime p as in Lemma 17: 1 1 pn � p < 2 pn ,
(
Then
)
1 n -1 n 1+_ n-1_ r P1 > p ---n > n
and by the induction hypothesis (181 ) But according to Lemma
17
n( n +1} 2k-2nk 2 r 2 Nk -r ( P1 ) Nk ( P ) � 2 P1 P
Weyl's sums
90
[ Ch. II, §
13
and, therefore, by (181)
Hence, since PP1
=
pn
2 n log (n + 1 ) we get
Respectively, under T > 3n log ( n + 1) we have e.,. < 2 ( n� 1) ' Hence it follows that for any e > 0 under k > 3n3 log (n + 1 ) and n � 21., we have the estimate n(n+l )
_ 2 k- -- +e 2 Nk ( P ) � 2 n k p 2 2
=
(
0 P
n(n+1» 2k- -- +e 2
)
•
(182)
On the other hand, by (171)
It is seen from comparison of this estimate and the estimate ( 182), that the order of the estimate ( 180) is almost best possible. A question about the least value of k, under which the estimate ( 180) is fulfilled, is much more difficult. This question is important in connection with the following circumstance: estimates of Weyl's sums obtained by the help of the mean value theorem are, as a rule, more precise, if one succeeds to establish an estimate of the form ( 1 80) under lesser values of k, i.e. , the lesser the better.
Ch. /I, § 13 )
Vinogradov's mean value theorem
91
Let us show that the estimate ( 183)
n ( n2+ 1 ) .
Indeed, according to (171) cannot be fulfilled under k < e, in order to satisfy the estimate (183), the estimate for re the
pk
=
0
(p2 k--n( n+2-l» +e)
Nk ( P)
�
Pk
and,
n( n+ 1 ) . 2
should be fulfilled, but that is possible only under k � Thus the best result which might be expected to obtain is getting a precise estimate (with respect to 1 the order) under k = ('; ) . The estimate (183) following from Theorem 15 was ob tained under k � 3n3 log (n + 1). Using the Linnik lemma (Lemma 9) instead of Lemma 7, we get now this estimate under k � 3n 2 1og (n + 1 ).
n +
p i1
LEM MA 18. Let n � 2, P � (2n) 2 n , p be a prime, � [pp -l] + 1 . Then under k � we have the estimate
n( n2+1 )
�
P
' n
where
N: ( AI P , . . . , A n p n ) is the number of solutions of the system
� l � " " " � �� � � l �
. . . . . xf + . - y� = A npn ..
},
. .
and the summation is over the region according to Lemma 16
1
::;;; Xi , Yi ::;;;
PPI + p ,
i f. j ::;. X i ¢ X j , Yi ¢ Yj ( mod p ) ,
IA"I
n 2 , then % l , . " , %k
p
=
and, therefore, " 1 Nk � 2' ( 2
n (n+ l ) 2 Nk - n ( PI ) . k)2n PI2 n 2k - -P
(189)
Now we obtain the lemma assertion from (185), ( 188), and ( 189)
n ( n + t) 2 Nk-n ( PI ) . Nk ( P ) � 2Nk, + 2Nk" � 2 ( 2 k ) 2n PI2n 2 k - -P
The recurrent inequality (184) enables us to make the statement of the mean value theorem essentially stronger, because this inequality reduces the estimation of t o the estimation for k Pd ( but not t o Nk- n 2 (Pd as i t was obtained earlier in Lemma 17).
N -n (
Nk ( P)
Vinogradov's mean value theorem
Ch. /I, § 1 3)
THEOREM 16. Let n � 2 , T � 0, k
eT
-
=
95
n (�+ I ) + nT, and
(
)
_ n(n - l ) I _ ! T n 2
•
Then for the number of solutions of the system ( 1 78) the estimate ( 1 90) holds under any P � 1 . Proof. Since, obviously,
then to prove the estimate ( 190) it suffices to show that ( 19 1 )
If T
=
0, then this estimate takes on the form
and i s fulfilled by ( 1 73) under any P � 1 . Apply the induction. Let under a certain T � 0 and k = n ( n2+ 1 ) + nT the estimate (191) be fulfilled under any P � 1 . Prove it for T + 1, i.e. , under k = n ( n2+ 1 ) + n ( T + 1) . We shall consider the cases P � ( 2n ) 2 n k 2 and P < ( 2n ) 2 n k 2 separately. If P � (2n ) 2n k2 , then by Lemma 18 Nk ( P )
� 2 ( 2 k) 2 n PI2n P2
where i p7i � P < p 7i and PI = [pp- I ] the induction hypothesis, we obtain I
1
+
n ( n+ 1 ) k- -2 Nk- n ( PI ) ,
1 . Since k - n
=
n( n + 1 ) + nT, then using 2
[ Ch. 1/, § 13
Weyt's sums
96
Observing that
P > 4 k 2 and, therefore,
P1 we get
< 2P
n(n+1)
(pP1 ) 2 k - -2 p{ r
1_1n + 1 < 2 P 1-1n ( 1 +
< 2" T p
n(n+1) 2 k - -- + 2
< 3 . 2" T P
2k-
1
2k '
( 1 - -1 ) € r n
n ( n+ 1 )
-2
)
( 1 + -1 ) 2 k - n 2k
+€r+l
But then it follows from ( 1 92 ) that
Let now
P
ko as well. It suffices to use the evident inequality
Nk (P ) � p 2 k - 2 k o Nko ( P ) and apply the estimate ( 190) to Nk o (P ) .
Estimates
Ch. 11. § 1 4)
of
Weyl's sums
97
Note . With the help of more complicated considerations [44] the mean value theo rem can be improved by removing the factor p e and so it is possible to get under n � 1 , k > cn 2 10g n, p � 1 the estimate ..
Nk (P)
k n(n+t) � C(n)P 2 - 2 ,
( 1 94 )
--
where c is an absolute constant and C(n) is a constant depending on n only. An elementary proof of the mean value theorem in the form ( 1 90) is obtained in the article [37] . §
14 .
Estimates of Weyl's sums
To obtain estimates of Weyl's sums by the Vinogradov method besides the mean value theorem we need two comparatively simple lemmas. L EMMA 1 9 . Let f(x) be an arbitrary function taking on real values. Then under any
positive integers P, PI , a,
an d
k we have:
P1 - l P i I(x) 21f e L L e21fi I(x+y) + P1 1 � � L P I '" O x=1 y= x=1 P P1 1 P 1fi I(x) 2 2 1fi I( x+ayz) + 2 aPl , e L 21 L L e p x=1 x=1 y,z =1 k+t 2 2k P P-l P i i k+1 /(x) /(x+y) 21f 2 21f �2 L e Le x=1 y= O :1: =1 P
10 .
-
2° .
,
�
3° .
L
Proof. Under any integer y � 0
y
P
L e 21fi 1(:1:) x=1
=
=
e21f i 1(:1:) +
y+ P
e 2 1fi 1(:1:)
L L :1:=1 :I:=y+l LP e21fi I(z+y) + 20yY,
( 1 95)
P+y
-
e 21f i 1 (:1:) L z=P+l ( 1 96)
:1: = 1
where I Oy l
� 1 . Hence, carrying out the summation over P P /(x) � L e21f i /( :I: +Y) 2 , 21fi e L + y z=1 z=1 PI
P
L e 2 1ri 1 ( :1: ) :1: = 1
Pl - l P
� yL=o
i
e21f I( :I: + Y) L :1:=1
y,
we get the assertion 1 ° :
+ PI (PI - 1 ) .
Weyl's sums
98
To prove the estimate summing with respect to
[ Ch. II, §
2 ° we replace y by ayz in the equality (196) and carry out y and z:
I: e27ri f ( x ) = L e 27ri f( x +a yz ) + 20(y, z) ayz, P
P
x=1
x=1
P; I: e27ri f ( x ) = I: I: e27ri f( x +a yz ) + 2a I: O(y, z) yz . x=1 y,z=1 x=1 y,z=1 Hence, because IO(y, z) 1 ::::; 1 and P
P
the assertion
PI
PI
2 ° follows:
x=1 y,z=1 x=1 Determine SI and PI with the help of the equalities 2k PI mm. ( [SI2 k� l ] + 1 , P ) . SI I: I: e27ri f( x + y) y= o x=1 P
P-l
=
=
1
SI2 k+ l ::::; PI ::::; P and, therefore, __
Then
1- " e27ri f( x + y) ( 1 S -21k S 2 k1+ l " "" P1 1 ) "" 1 ' p1 y=O L.J x=1 L.J Hence, using the estimate 1 we get the assertion 3°: P
1
I
P
__
�
�
_
0,
I ::::; ;1 L L e27ri f ( x + y) + PI - 1 ::::; 2 S12k+ 1 , x=1 y= o x=1 2 k+ l 2k L e27ri f( x ) ::::; 22k+ 1 SI 22 k+ 1 I: I: e27ri f( x + y) x=1 y=O x=1 P
14
I: e 27ri f ( x )
PI - l
P
P
__
P- l
=
P
Ch. 1/, § 1 4 ]
L EMMA
Estimates of Weyl's sums
20.
If a function
99
F(a l , ' " , a n ) is given by the multiple Fourier expansion 00
L and
satisfying the condition
F ( a 1 , . . . , a n) � O, , qn we have th en under any positive integers Q1 ,
00
•
•
•
L
Proof. Since
F(a 1 , . . . , a n ) � 0, then F (a t , . . . , an )
�
91-1 '" I=O XL
9n -1 F (a 1 + X l , . . . , a n + X n ) L Qn Q1 x n =O
( 1 97)
00
=
L
By Lemma
X l=O x n =O 2
Using this equality, we obtain the lemma assertion from ( 197 ) : F ( at ,
.
. . , an )
00
L 00
L COROLLARY . Let f(x)
= al X + . . . + a n x n and S (a 1 , . . . , a n ) � n( n+1)
Then under any positive in tegers r
n
e2 11" i /(x ) . L x =l p
=
and k we have the estimate
I S(al , . . . , a n ) 1 2 k � k n - 1 P-2- -r
L N�P\O, . .
i'>'. i < k p r
.
, }. r ,
.
.
.
, 0) e211"iar.>.r .
Weyl's sums
100
[Ch. II, §
14
Proof . Let us consider the function
F(
a t . . . . , a n) IS( at. . . . , a n) 1 2k . =
By ( 1 59)
where the range of summation is
IA " I
( v = I , 2, . . . , n ) . kP" Since F( � 0 , the lemma conditions are satisfied. Choose I if v = r , = if v t= r . kP" Then using the lemma we get
aI , . . . , an )
' 1 , , >' n) Al , ... ,An •
•
•
>' n ) e 27ri (A I Pl(y) + . .. + AnPn(Y» P-l
L e 27r i (A I PI (y) + .. . + An Pn(Y» ,
y=O
[ Ch. II, § 14
Weyl's sums
108
where the range of summation is
( = 1 , 2, . . . , n). v
Hence, using the Cauchy inequality (1 43 ) and the relation (165), we get
4k + 2 P e271'i /(':) L .:=1
2 P-1 e271'i (AIPI ( y )+ ... +AnPn (y» L y =O (212)
where
P-1
V(P)
=
L L AI , ... ,An y= O
e271'i (AIPI (y)+ ... +AnPn(y»
2 •
Now we shall estimate the magnitude of V(P). Observing that f3n ( Y ) does not depend on y , we obtain V(P) � 2 k p n
.; 2 k P '
P-1 L
,�o � (
� (2 kt p n
m;n 2 kP ' ,
( Ly, z 1 + Ly, z � '
2I 1 p.(y)
� ,1, ( ' ) 11 )
(213)
where the sum
is extended over those values of y and inequalities
z,
which under a certain t � 1 satisfy the
( s = 1 , 2 , . . . , n - 1). Respectively, the sum
(214)
Estimates of Weyl's sums
Ch. /I, § 1 4] is
over those values of y and z, for which there is
s
109
(1 �
s
� n - 1 ) such that (215)
By Lemma 23 for the sum E 1 the estimate n(n- 1 )
L 1 � (2 n) 3 n P -2- + 1 t y,%
holds. Applying the estimate (215) for one of factors in (214� and estimating all the other factors trivially, we get
2: 2 � y , ::
P-1
2:
P
y , z= O
n(n - 1 ) 1 n(n-1) 2 - = -1 p 2 + 2 t t -
•
-
Since by virtue of (213) V(P) � (2k) n p n
(L y,%
)
1+L2 ' y , ::
then choosing t = [ vIP] + 1 we obtain
(
V(P) � (2kt p n (2n) 3 n P � 3 (2k) n (2n) 3 n p
n(n- 1 ) -- + 1 VP 2 + 2
n(n +t) 3 2 - + '2 .
)p P -2- +2 ) n(n-1)
Substituting this estimate into (212) , we get P
L e 211'i /( x )
x=1
4 k+ 2
� 3 (2kt2 4 k + 2 (2n) 3 n N2 k (P)P
[i + � n log n] + 1 , n(n + 1 ) k = [� + ] + nrl ' 4
Choose
ri =
n(n + 1 ) 3 -2 + '2
r = 2r1 ,
It is easy to verify that the estimates
2k
� ?'
n(n + 1 ) + nr, 2
r > 3 n log n + n,
(216)
110
Weyl 's sums
(
)
r n(n - 1 ) n(n - 1) 1 _ .!. < n �� 2 hold. Therefore, using Theorem 16, we obtain
[ Ch. II, §
. 1 , . . . , A n ) L e2 11'i (a l ,x1 x + ... +a n ,x n x ) x=1 >'l , ... ,>' n
2k
(23 1 )
and the summation is extended over the region
1J.l ,, 1 < kP " Proof.
(v = 1 , 2, . . . , n ) .
Using the inequality ( 142) , we obtain
I S I 2k �
t e2 11' i (alxy+ . . .+an x n yn ) )
(tx=1 y=1
2k
P P 211'i (a1x y+ ... +a x n y n ) 2k 1 p2ke n � L L x=1 y=1 Since by (159) P
e 2 11'i (a1 xy+ . . . +a n x n y n ) L y=1 =
2k
(232)
[ Ch. II, §
Weyl's sums
120
where the summation is extended over the region
I A1 1
2 and f ( x ) a l x + . . . + a n xn be a polynomial with integral coefficients . Consider the rational exponential sum
P
If ( an, q) = 1
.
f( x )
q . S(P) = I:> 2 71" x=1 and q = pr, then under 1 � � n - 1 the estimate r
1 - --"---
I S ( P) I � e3 n P 9 n 2 1og n follows from Theorem 17. Using the repeated application of the mean value theorem, this result can be slightly strengthened. So it follows from Theorem 1 9 that for a certain interval of values we have the estimate 1 - 2.. ( 2 42 ) I S ( P) I � CP n 2 , . where C and 'Y are absolute constants. Under an arbitrary positive integer q the estimate (242) is the best among known ones and no approaches to the problem of its essential improvement are seen for the present. But this estimate can be strengthened under a special choice of the denominator q. Let us show how it can be done under q being equal to a power of.a prime. LEMMA 28 . Let (x, n, P be positive integers, p > n 2 be a prime, F (x) bo + b 1 x + . . . + bn xn , and Ta (F, P] be the number of solutions of the congruence F( x) == 0 ( mod pa ), O � x < P. ( 24 3 ) r
=
If (bo , . . . , bn , p) = 1 and
holds.
P � pn , a
then the estimate
a Ta [F, P] � 2n Pp - 2 n
=
Proof. At first we shall show that under P ap8 with 1 � a < p and � 0 we have the estimate Ta (F, ap8] � nap 2 n . (244) Under = 0 this estimate is trivial, because a p n � P = a < p, s
s
Or 8- -
Incomplete rational sums
Ch. /I, § 1 7]
and, therefore, nap - 2 n '"
>
n,
127
but the congruence o�x
0',
In fact, if 8 � r, then we arrive at a contradiction: 4 n2
� 0' < 4 ( r + 1)(8 + 1) � 4 (r + 1 ) 2 < 4 11. 2 ,
and, therefore, 8 � r + 1 . Further, it is obvious 8(4r
+ 8) � 4r8 + 48 + 4 ( r + 1 ) = 4(r + 1)(8 + 1 )
>
0' .
�
Finally, P > p r+ 1 � p4 s . In Lemma 19 we choose PI
= a =
. f(x )
P
pS .
Then we obtain p'
P
1 L e 271" -q- � 2s L
P
x=l
Le
2
x=l y , z= 1
' f(x +p ' y z) p o. + 2p3 s . 71'1
Denote by M a set of those x from the interval 1 � single congruence of the form
x
� P, which do not satisfy a
2r + 3
Po (e) in such a way that under n = [ ;] + 1 the estimate p
L e21fi m/( x) 1� m � n 2 x=l max
7r
( 27 0 )
eP � 4 ( 1 + 2 10g n )
would be satisfied. Then, using the expansion of the function .,pI ( { f( x) } ) into the Fourier series, we obtain
p
L .,pl ({f(x)} ) - 'Y P x=l
=
m= - oo p e2 1fi m/ (x ) ' C (m e ) P L L 1 + x=l m =-oo 00
+ L I C1 (m) 1
Iml>n 2
p
L e21fi m J (x) x=l
=
(the sign in the sum indicates the deletion of the summand with m 0). Hence applying the estimate (270) under I ml � n 2 and the trivial one under I ml > n 2 , by virtue of the lemma we get I
p L.J '" .,pl ( {f( x) } ) - 'YP
x=l
� �
� eP + 4 + P L.J e P L.J ' 7r 2 2 ( 1 + 2 log n) ml Iml > n 2. 7r m e m= -n 2 l 2P 1 - < 2eP. eP + - e P + -2 2 7r n 2 e 7r
�
",
1
1
Fractional parts distribution
144
[ Ch. 11/, §
19
In the same way the estimate p
L 7fJ ({ f(X) } ) - "IP x =1 2
:::;; 2eP
is obtained. But then it follows from (269) that
-2eP :::;; Np C'Y) - "I P :::;; 2eP
and, because
e
can be
small as we please, we get
as
lim N ) = "I. P -oo p pC'Y
I
The sufficiency of the condition (267) is proved. Now we shall prove the necessity of that condition. Indeed, let the function f( x) be uniformly distributed. Take m f= 0 and choose an integer q > Im l . Denote by Mk a set of those x from the interval [1 , Pj , which satisfy
�q :::;; { f (x) }
': P 21r i a /(x) e L x=1
11=0
x=1
h
Substituting this estimate into (274), we get the assertion of the lemma: P
L e21ri /( x )
2
x=1
� �
THEOREM 25 .
P1-1 2P (PP1-1 2 p2
21ri a le x ) e L � h < Pl 21r i � /( x ) + 2P1 + max t e l �h < Pl x= 1
+ 2P PI + 2P l max
1
P
x=1
h
)
+ 2 pl .
A sufficient condition of uniform distribution of a function f(x) is unifonn distribu tion of its finite difference � f( x) under any integer h � 1 . h
Proof. Let according to the hypothesis of the theorem fractional parts { � f( x ) } be h
uniformly distributed under any positive integer k. Then by the Weyl criterion under eve ry integer m 1= 0
P 21r m a / ( x ) "e i P-+oo P xL.J lim
1
-
=1
h
=
(276)
O.
Apply the inequality of Lemma 30 to the function m f( x ) Then observing that we obtain
Il m f ( x ) = m � f(x)
.
h
h
1 (
P
2 P 21r i m a / (X » P L e 21ri m / ( x ) � 2P P P1- 1 + 2 1 + max L e l �h < Pl x= 1
x=1
h
P
)
•
(27 7)
Let 0 < e < 1 and P1 = [e;62 ] + 1 . If follows from (276) that 2 = P2 (e, m) can be chosen in such a way that under P � max (Pt , P2 ) the inequality
P
Le
x= 1
21r i m a / ( x ) h
Fractional parts distribution
148
t
111.
§ 19
P � max ( !�P1 , P2 ) ' Then we get from (277) 2 e 2 71'i m f ( x ) � 2 P e 2 P + e 2 P + e 2 p e2 p2 ,
will be satisfied. Choose
x=1 p
[ Ch.
(�
'"' e 2 71'i m f ( x )
L.J
., = 1
�
�
)
=
/ """ e P,
and, therefore, 1 P-oo P lim
-
p
.
'"' e 2 mf( ) 71'J .,
L.J
x=1
=
O.
The theorem is proved by virtue of the Weyl criterion. THEOREM 26 ( Weyl's theorem ) . If a polynomial
f(x)
=
a o + at X + . . . + a n x n
(278)
bas at least one non constant term witb an irrational coefficient, tben its fraction al parts are uniformly distributed. Proof. We shall start with a case when the coefficient of the highest degree term is irrational. Under n = 1 the polynomial (278) is in reality a linear function ao + at x with an irrational coefficient at . By (273) fractional parts of such linear functions are uniformly distributed. Apply induction. Let n � 2 and the theorem be proved for polynomials of degree n - 1 , having an irrational coefficient of the highest degr()e term. Choose an arbitrary positive integer h and consider the finite difference
l:::. f(x) h
=
f(x + h ) - f(x)
=
a n [ (x + h ) n - x n ] + . . . + a d(x + h) - xl .
Evidently, l:::. f( x ) is a polynomial of the ( n - l ) -th degree with an irrational coefficient h of the highest degree term. By the induction hypothesis, fractional parts of this polynomial are uniformly distributed. But then by Theorem 25 fractional parts of the initial polynomial are unifonnly distributed also. Thus the theorem is proved for polynomials with the leading coefficient being irrational. Now let 1 � s < n and as be the leading among irrational coefficients of the polynomial f(x) . Denote by q the common denominator of coefficients as + h ' " , a n and write the polynomial (278) in the form
f(x)
=
(x) h e x) +
n too ) . The theorem is proved completely. By Theorem 27 there exist functions f(x) such that the system of functions f( x + 1 ) , . . . , f(x + s ) under s, which does not exceed a certain bound, is uniformly distributed in the s-dimensional unit cube . In the following theorem it is shown that there exist functions for which the restriction on the magnitude of s may be lifted. A function f (x) is called completely uniformly distributed, if for any s � 1 the system of functions (287) f(x + 1 ) , . . . , f(x + s ) is uniformly distributed in the s-dimensional unit cube. It follows from (280) that a function f( x) is completely uniformly distributed if and only if under every s � 1 and any choice of integers l , not all zero the function
m . . . , ms F(x) m t f ( x + 1 ) + . . . + ms f ( x + s ) =
is uniformly distributed. THEOREM
28.
Under any
a
>
4
a
function
f(x) is completely uniformly distributed.
00
=
f(x)
determined by the series
L e - k a xk
k=o
(288)
Fractional parts distribution
154
[ Ch. III, § 20
be arbitrary integers not all zero and the function F (x) be Proof. Let determined by the equality (288). Under n � 2 s we determine Q ( x) and R(x) wi th the help of the equalities
m}, . . . ,m .
n
Q( x) L i} k x k , k=O
R(x)
=
where
i} k = e -k "' .
=
L
k=n + l
Further, let
Qs( x) = m l Q( x + 1 ) + Rs( x) = m lR(x + 1) + Then, evidently,
00
. + ms Q( x + s ) , . . . + msR(x + s ) . .
.
f (x) = Q (x) + R( x) and F( x)
=
ml (Q( x + 1) + R( x + 1))
+ m . (Q( x + s ) + R(x + s ) )
+... =
Q.( x) + R. (x).
B y virtue of the multidimensional criterion of Weyl i n order t o prove the theorem it suffices to show that under any fixed positive integer s the estimate p
L e2 x=l
71'i F (x) o(P) =
is satisfied. Using Lemma 26, we get p
p
71'i F(x) L e2 x=l �
71'i ( Q . (x) + R.(x)) L e2 x=l p p L e2 71'i Q . (x) + 27l' L IRs( x) l · x=l x=l
At first we shall estimate the magnitude of p
R L I Rs( x) l · x=l =
Determine n from the condition nO'- l � log
P < (n + 1 t-1
(289)
Uniform distribution of functions systems
Ch. III, § 20]
and choose P in such a way that the inequality m = max1 � 1I � " Imll l, is satisfied . Then we obtain
..
P
155
n >
..
max ( 4ms,
20' +1 ) ,
where
P
R = L L m Il R( x + v ) :::; L l m Il I L R( x + v ) x= 1 11=1 11=1 x=1 R P :::; sm L R( x + s) = sm L e-k" L(x + s)k . x=1 x=1 k=n+1 00
Hence, because of
k+1
�( x + s ) k � (x + s + l ) k+ l (x + s)kH
L...J
�
�
x=1
it follows that
L...J x=1
( P + s + l ) k+ l k+1 '
max (4 ms, 2u + 1 ), then we have the estimates
1
= r.
(292) Since 0 :::;;
t
1 - --:=-- 2 4 n2 10g n
4 and n a -1 � log P < ( n + 1) a- 1 , then _
e3 n
P
as P � 00 , and, therefore, estimate
p
S
1
2 4 n 2 10g n
<e
3n- � 2 4 log n
---+
0
,
= o(P). But then by (289) and (291) we obtain the
L e 2 ". i F( x ) � l S I + 21rIRI
x= 1 equivalent to the theorem assertion.
=
o(P)
[ Ch. III, § 20
Fractional parts distribution
158
Note. If a function I(x) is completely uniformly distributed, then under any choice of positive integers t and r the system of functions
(296)
I(tx + 1 ), . . . , /(tx + r)
is uniformly distributed in the r-dimensional unit cube. Indeed, let m 1 , ' . . , mr be arbitrary integers not all zero. To prove uniform dist ri bution of the system of functions (296) by the multidimensional criterion of Weyl it suffices to show that the sum P
S = l::: e 2 11" i F( t x ) ,
x=1 m d ( x + 1 ) + . . . + mr/(x + r) , has a nontrivial estimate S = o(P).
where F(x) = Using Lemma 2, we obtain S=
x=1
1
t l::: a=1 t t P 211" . 1 � l::: l::: e 1 I SI t
tP l::: e 211"i F ( x ) Ot (x)
=
t
(
)
t P 2 . F (x ) + ax T , l::: e 7r 1 x=1
( F(x ) + -axt ) .
a=1 x=1 Determine a function Fa (x) by the equality � Fa ( x) its finite difference with step h :
(297)
Fa (x) = F(x) + at"' and denote by
h
� Fa (x) =
h
ah Fa (x + h) - Fa (x) = F(x + h) - F(x) + - . t
The difference F( x + h) - F( x) is, obviously, a linear combination of consecutive values of the function I(x) :
F(x + h ) - F(x) = m 1 ( J (x + 1 + h ) - / (x + 1 )) + . . . + mr (J(x + r + h ) - I(x + ) ) = m U (x + 1 ) + . . . + m � + hl (x + r + h ) , r
where m � , . . . , m �+h are integers not all zero. Hence, because the function I( x) is completely uniformly distributed, by (288) the function F( x + h) - F( x) is uniformly distributed. At the same time the function � Fa (x), which differs from F(x + h)-F(x)
h
by an additive constant only, is uniformly distributed as well. But then by Theorem 25 the function Fa (x) is uniformly distributed too. Therefore, under any a from the interval 1 � a � t we have
Ch. /II, § 21 1
Normal and conjunctly normal numbers
an d it follows from (297) that
� L: t
l SI �
The assertion (296) is proved.
159
tP
L: e 2 1ri F. ( x )
a=1 x = 1
=
o(P) .
§ 2 1 . Normal and conj unctly normal numbers
Let q � 2 be an integer and a be an arbitrary number frorp the interval (0 , 1). Let us write a by means of its q-adic expansion a =
0' 'Y1 'Y2 ' ' · 'Yx · · · .
(298)
Denote by N ( P) (0 1 . . . O n ) the number of satisfactions of the equality (299) (x = 0, 1 , , , . , P - 1 ) , 'Yx + 1 . 'Yx+ n = 01 . " On where 01 . . . On is an arbitrary fixed block of digits 0" E [0, q - l] and the equality (299) is considered as the equality of integers written by means of their q -adic expansion. As in § 8 , N ( P ) (0 1 . . . o n ) is equal, evidently, to the number of occurrences of the given block 0 1 . . . On of digits of length n among the first P blocks "
'Y1 " ' 'Yn , 'Y2 · , , 'YnH ' . . . , 'Y p . . · 'Y P +n- 1 formed by successive digits of the q-adic expansion (298) for a . The number a is called normal to the base q, if for any fixed n � 1 under P -+ 00 the asymptotic equality 1 N ( P ) ( 01 . . . On ) = -n P + o(P) q holds. The theory of normal numbers is closely connected with problems of uniform distri bution of fractional parts of exponential functions aqx . The following general lemma about uniform distribution of fractional parts of an arbitrary function f(x) lies at the foundation of this connection.
LEMMA 3 1 . If there exists an infinite sequence of positive integers m 1 < m2 < . . . < m n < . . . such that under every n � 1 and any integer v with 0 � v � m n - 1 the number T" of satisfactions of the inequality v v+l ( = 1 , 2, . . . , P) � { J( x) } < mn mn satisfies 1 T" = - P + o(P) ( 30 0) mn as P -+ 00 , then fractional parts of the function f( x) are uniformly distributed. -
x
Fractional parts distribution
160
Proof. Choose an arbitrary (3 parts {f(x) } (x = 1 , 2, . . . , P) with the help of inequalities
[ Ch. III, § 21
(0, 1] and denote by Np «(3) the number of f act i on l falling into the interval [0, (3). Determine an integer b
E
r
a
Then, obviously,
Using the condition (300), we obtain
( mbn ) = vL6 -=1o Tv = -m n P + o(P), ) = L6 Tv = P o(P) N ( Np p
b
-
b+l
b+ l
-:;;;-
-:;;;-
n
v=o
n
+
aqd, therefore,
(�n (3) p -
+ o(P) �
I Np «(3) - (3p l
�
Np «(3)
-
(3P �
1
C:n1 - (3) p
m n P + o(P).
INp «(3) - (3p l
�
no = no(e) so that for n � no
�P + o(P).
Po = Po(e) we ob tain I Np «(3) - (3pl
and, therefore,
o(P),
-
Now let an arbitrarily small e > 0 b e given. Choose the inequality _1 _ < £2 is satisfied. Then, evidently, mn
Hence under P �
+
I p-+oo p Np «(3 ) lim
�
eP
=
(3,
which is identical with the lemma assertion. THEOREM
29 . A nwnber a is normal to the base the function aq X are uniformly distribu ted.
q
if and only if fractional parts of
Normal and conjunctly normal numbers
Ch. 11/, § 21 J
Proof. Choose an arbitrary block 151 . " t5n of digits with 0 � OJ � an integer v with the help of the equality
Let under a certain
x
161
q - 1 and determine
the equality (30 1 )
be fulfilled. Then
and, therefore, the inequality v
-n �
q
{ aq x }
m t + e ) is investi gated rather easily with the help of the following lemma. �
(3"
m1 m 1 7 m71 .
71
.
m, ml, 7, 71 be determined according to (3 1 8 ) , d (b, m). Then under d :::1 [or every
LEMMA 32 . Let q � 2, the quantities b be an arbitrary positive integer, and P � we have the estimate
7
0
T mml TI -
::;;;
=
vm . the estimate
=
ym .
TI < PI . . . Ps
::;;;
Viii
r=
we obtain from (326)
I R I ::;;; C{e )m� + · + vm ( 1 + log m) 2 = o (m� +·) . Now it follows from (324) that
T!:) (b , h)
=
� P + o(m�+e) .
Hence by (322) and (323) we get the assertion of the theorem:
N!:) ( D J . . . Dn )
=�
[; + { :� }] P + R = q� P + o (m i +E) .
Note that the uniformity of distribution of digit blocks DJ . . . Dn in a part of the period of the fraction ! follows from Theorem 32 only if P belongs to the interval
m2' + E < P < 1
T, i.e. , if the period is sufficiently large and a sufficiently large part of it is considered. It is so, for example, if we require the fulfilment of the inequalities > 2(3" ( II = 1 , 2, . . . , s ) . Since = and T = � , Indeed, in this case by (318) then
a"
mJ = pfl . . . p�' .
� !?!:..
T mJ Let a maxJ �"�s a" . Then PI
m 01 + J � . � . . -PI I -P. �. � p p p
=
"
p�1 . . . p�. o. J . P 2+ s
.
(327)
=
.
. •
Ps � (pfl
"
.
p
I = m aI
� )'x .
and, using the estimate (327), we obtain the required bound for the magnitude of the period:
T � (mp . . . Ps) 2 i
!.
�
.. 20 . m !.2 + ...!.
Now we start on the question concerning the distribution of digit blocks in a small part of the period. This question is more difficult than the former and more com plicated methods of the estimation of corresponding exponential sums have to be invoked for its solution. We restrict ourselves to the case where P > 2 is a prime. We assume that the quantities and (3 are chosen, as before, by (318).
T, Tt ,
m = pO,
Distribution of digits
Ch. III, § 22]
171
33. Let (q, p) 1, (a, p) = 1 , > 1 6 ,8, and r r p OI . If 2 � r < 8P ' then we have the estima.te equality p
THEOREM
=
a
be determined by the
=
where
2 . :06 ' Proof. If P � e 3 6 r , then the theorem assertion is trivial, because of 'Y
=
1 - .1. 36 ')' 3 P r2 � 3 Pe - -r- � 3 Pe - 1 8 ,), > P,' Let
P > e 3 6 r . We determine integers s and n with the help of the conditions s�
a
4r
< s + 1,
n
a
< - � n + 1. s
(328)
It i s easy t o verify, that the estimates
s
>
7 � n < s (p - l )
(3 ,
holds. In fact, since by the hypothesis
s
a
> 4r
-1
Further, evidently, n � ;- - 1
a
n
a
>
8,8r and pOI
=
p r , then we obtain from (328)
2{3 - 1 � ,8,
� 4r - 1 � 7 and, finally,
log p 1 � 6 r < -6 log P = -6- < s log p < s (p - 1 ) . a
s Determine integers aI , . . . , a n by means of the equality r
(329) and show that under ( u , p) = 1 there are no multiples of p8 among the quantities " all (v = 1 , 2, . . . , n). Indeed, by comparing the coefficients of x , we get
_ I" n! II ( mod p8 ) . v.
all = Let pW. be the highest power of p dividing obtain
W II �
[�]p + [ p� ]
+
;t .
u
Then because of n < s (p - 1 ) we
n p-l
. . . < __ < s ,
Fractional parts distribution
172
and, therefore, Denote by
Ts
( a " , p S ) = Pw. ,
o
� W"
(330)
{3, then by (96)
TI P
s-p
r £ we have the estimate
aq 2:: 2 1Ti '"
P- l
e
x=O
p c.
2 is a pz:ime, ( a , p) = 1 , = pOi we have the equality If
2 .:0 6 '
Proof. We determine integers t, b, and
Then we obtain from
(q , p) = 1 ,
p r = pOi, and 3 � r < 8P '
h by the equalities (32 2 ):
(323) and (324) h N ( P ) ( Cl . . . cn ) = - P + R , m
m
(338)
Connection with quadra ture formulas
Ch. III, § 22)
where by
175
(325) the estimate
holds. Hence it follows that
0'- 1
I RI � L
L
v =Lp v=O
Le
2 11" i
azqz pO
--
.:= 0
z
v=O ( z,p O )=p. , l �z <po
0'- 1 1
P-l
1
-
L
(zl ,p)=I , l � Zl <po - .
Since by the assumption p r = p o' and 1, 2, . . . - 1 ) , we get under v � �
,a
3 � r < ;'p , then choosing r v = r a � v
(v
=
a-v
2 � rJJ < � and by Theorem
32 the estimate
.:= 0 ,
holds. Using this estimate and applying the trivial estimation under v � � we obtain
Now, observing that
m
0( 1 ) , h=qn +
we get the theorem assertion from
(338)
[ Ch. III, § 23
Fractional parts distribution
176
§ 23. Co nnection between exp onential sums , quadrat ure formulas and fractional parts distribution
As it was noted in the introduction, there exists a close connection between est i mates of exponential sums and approximate calculation of multiple integrals
J . . . J !(Xl , . . . , x s ) dxl . . . dx s . 1
1
0
o
This connection is established especially simply, if the function ! ( Xl , . . . , x s ) has period 1 with respect to every variable Xl , . . . , X s and the Fourier expansion 00
(33 9)
ml ,,,·,m,=-oo converges absolutely. Consider a quadrature formula
where -Rp ( f] stands for the error obtained in replacing the integral by the arithmetic mean of the integrand values calculated at the points
( k = 1 , 2, . . . , P). The set of points Mk is called a net , and the points are said to be nodes of the quadrature formula. Let a certain system of uniformly distributed functions It ( x) , . . . ' !s ( x) be given. Then under any choice of quantities "Iv E (0, 1] ( v = 1, 2, . . . , s ) the number of fulfilments of the inequalities
o � {fl ( k ) }
< "11 , . . .
, 0 � { fs ( k) }
1
L'
nl "
then, in
.. , n , = - oo
(�)a� a-I
1 we shall use the notation
n + +n f n 1 , ... , n . ( X l , . . . , Xs ) - a i a... n l. f ( xa n . x s ) Xl Xs > 2 and function f (x1 , ' ' ' ' xs ) belongs to the class E� ( C ) . t , o o . ,
_
•
LEMMA 33 . Let Q we have the equality
•
•
a
f (x t , . . . , Xs ) =
1
1
L
1
. . . J f Tl t ... ,T. ( Y1 , . . . , Ys ) J Tl ,,,.,r, =O o
0
Then
Quadrature and interpolation formulas
Ch. 11/, § 24]
)
(
T II { Yv - x v } - 2"1 . dY1 . . . dys . s
X
197
v= l
(387)
Proof. At first we observe that under a > 2 the fact, that the function f(x l , . . . , x 8 ) belongs to the class E'; ( C), implies the existence and continuity of the derivatives
(
f Tl .... ,T· ( X 1 , " " x ) s
TV = 0, 1 , v = I , 2, .
.
. , s).
Let s = 1 , a > 2, and f(x) E Ef(C). Performing the integration by parts and using the periodicity of the integrands, we obtain
J f' ( y) ( {y - X } - � ) dY = J f'(X + Y) (Y - �) dY 1 1 (y - �) f( x + y) \ o J f( x + y ) dy f(x) J f(y) dy , 1
1
0
o
1
=
=
-
o
-
0
and, therefore,
Applying this equality to the variables X l , assertion:
•
•
•
, X 8 consecutively, we get the lemma
Note. If r is a positive integer, a > r + 1 , and f equality analogous to the equality (387):
E
E'; (C), then we have the following
Fractional parts distribution
198
where Br(x) are the Bernoulli polynomials:
Bt (x)
=x
1 2'
B2(x) .= x 2
-x
[ Ch. 11/, § 24
1
+ 6'
Under r = 1 this assertion coincides with (387), and in the general case i t i s proved by induction with respect to r with the use of the equalities ( r � 2).
THEOREM
39 . Let r � 2 be a positive integer, a � 2 r , and a I , . . . , a . be optimal coefficien ts modulo p. If a function I (XI , . " , x. ) belongs to the class E�(C), then we have the equality I (x t , . . . , x . )
+o
( r) log'Y p
p
(388)
'
whe:e a constant 'Y depends on r and s only. Proof. Let functions It ( X l ! . . . , x.) and 12 (x 1 , . . . , x.) belong to the classes E� ( Ct ) and E�(C2 )' respectively. We shall show that the product of these functions
fa (X I , ' " , x . ) = It (XI , ' " , x . ) h ( xI , ' " , x. )
belongs to the class E�(C3 ) ' where 03 depends on OI , 02 , a , and s . Indeed, denote b y OJ(mI , ' . . , m . ) (j = 1 , 2, 3) the Fourier coefficients of the func tions 11 , 12 , and fa . Multiplying the Fourier series of the functions It and 12 , we obtain 00
ml ''''J m , = -oo
where
C3 (m 1 , . . · , m 8 )
e 2 '11" i (mlxl+ .. . +m . x .)
,
00
n l , . . . , n . = - oo
Therefore, 00
f=
n l " .. , n . = - 00
�
=
01 02 . m . . n . ( l - nl ) . . . ( ma n l o . . . , n . = - oo [n l 01 02CT(mt ) . . . CT(m. ), _
-
n.
)]
0/
(389)
Quadrature and interpolation formulas
Ch. /II, § 24]
199
where u(m) denotes the sum
> 1,
Estimate the sum u(m). If m
u( m)
=
then
L [_ ( m 1- n ) ] a + L l2 m lm
2
Inl� l l
n
Inl> l l
1
[ ( m - )] a _
n
n
1
(m - n t 1
00
L
n = - oo
fi a
This estimate is, evidently, satisfied under m
=
1
too. But then we get from
(389)
According to the note of Lemma 33 under
(390) the equality
f( x I , . . . , Xs ) =
1
L 1"1 , · · · , r, = 0
J . . . J F( Yl , . . . , ys ) dYl . . . dys 1
1
o
0
holds. Differentiating the Fourier series 00
m l ,. · · , m .. = - oo
C(m 1 , · . . , s ) e2 1r i ( m l Y l + ... + m , y. ) , m
(391)
Fractional parts distribution
200
[ Ch. III. § 24
we obtain
j rTl , . . . , rT, (y l , · . . , Y
..
)
00
= C'
m l , I I . , m , = - oo
where
C' = (21rir ( Tl + . . . + T, ) .
m r1 Tl
" '
mr.. T' C (m 1 , . , m s ) . .
e 2 11" i ( m l Y l + . . . + m , y, )
,
Since
T r T, I ml I T l . . . I m .. I rTI . rT . C m C . m .. ' ml , . � , m ( . · 1 l )1 ( ml · · · m .. ) a C � C � a r ( ml . . . m .. ) r ' ( ml . . . m.. ) ..
.....,
.....,
the function h (Y l , " " Ys ) = rTI , . ..,rr' (Yl , " " Y ") belongs to the class E�(Cl ) with the constant C1 = IC' I C. Let c ( m) be the Fourier coefficients of the r-th Bernoulli polynomial Br( { y } ) . Since c ( m) = O( �. ) , then for the Fourier coefficients of the function
h(Y l , ' ' ' ' Y'' )
IT B;" ( {YII - XII } ) ..
=
11 =
1
we obtain the estimate
and, therefore, the function h ( Y1 , ' . . , Ys ) belongs to the class E; ( O2 ) . But then the function F ( Y l , . . . , y .. ) determined by the equality (390) belongs to a certain class E;(C3 ) and for the evaluation of the integrals in the equality (391) we may use the quadrature formula obtained under P = p in Theorem 38:
/ / F(Yl , . . . , y.. ) dYl . . . dys 1
1
. . .
o
0
where 'Y depends on r and
8
only. Hence by (390) we have the equality
Ch. III, § 24]
Quadra ture and interpolation formulas
201
which coincides with the theorem assertion by the definition of the function
F(Yl , " " Ys ) .
The interpolation formula (388) is obtained under the assumption that the function
f ( x l ! ' " , xs ) belongs to the class E� (C), where � 2 r and r � 2. In the same way, somewhat complicating the proof, we can convince ourselves of the validity of the formula under r = 1 also. So if f( X l , ' " ) E E� ( C) and al l " , as are optimal Q
coefficients modulo
, xs
p,
.
then under P = p we have the equality
(392) where 'Y depends on s only. Unlike the formula (388), which is not unimprovable, the order of the error decrease in the interpolation formula (392) cannot be improved under any choice of nets. The quadrature and interpolation formulas with parallelepipedal nets established in this section were obtained under the assumption of the equality P = p, where P is the number of the net nodes and p is the modulus of the optimal coefficients. If the quantities at , . . . , as are chosen so that the numbers 1 , al , , a s are ( s + 1 ) -dimen sional optimal coefficients modulo p, then these formulas are valid under P < p too, but then their precision will be lowered. So, for example, in the formulas (369) and (388) the order of the error decrease will be not O ( lo�: P ) or O ( lo�"'r P ) but 0 ('°8; P ) only. The first results in the application of the number-theoretical nets to the approxi mate computation of integrals of an arbitrary multiplicity were obtained in the papers [23] and [29] . Henceforward an essential contribution to the number-theoretical meth ods of numerical integration was made in the articles [3] , [12] , [14] , [10] , [8] , and [5] . Recently a large number of papers and a series of monographs [30] , [15] , [18] , and [38] have dealt with number-theoretical methods in numerical analysis. .
.
.
REFERENCES
G.
I . ARHIPOV , Estimates for double trigonometrical sums of H. Weyl, Trudy Mat. Inst . Steklov, 142 (1976), pp. 46-66. ( In Russian. ) [2] K . 1 . B AB EN KO , Fundamentals of Numerical Analysis, Nauka, Moscow, 1986. (In Russian. ) [3] N . S . BAHVALOV , On approximate computation of m'ultiple in tegrals, Vestnik Moskov. Univ . , Ser. 1, Mat . Mech. , 4 (1959) , pp. 3-18. (In Russian. ) [4] D . B U RGESS , The distribution of quadratic residues and nonresidues, Mathe matika ( London ) , 4 (1957) , pp. 106-1 12. [5] V . A . BYKOVSKII , On precise order of the error of optimal cu bature formu las in spaces with dominan t derivative and quadratic discrepancies of nets, Preprint, Computing Centre of Far-Eastern Scientific Centre of the USSR Academy of Sciences, Vladivostok, 1985, No. 23, 31 p. (In Russian.) [6] K . CHANDRASEKHARAN , Arithmetical Functions, Springer-Verlag, Berlin,
[1]
1970.
J. VAN ' DE R CORPUT ,
Diophantische Ungleichungen, Acta Math. , 56 (1931), pp. 373-456. [8] N . M . D O B ROVOLS KII , Estimates of discrepancies of generalized parallelepipedal nets, Tula Pedagogical Institute, Tula, 1984, dep. in VINITI 1 7 01 1985, No 6089. (In Russian. ) [9] T . ESTERMA N N , On the sign of the Gaussian sums, J . London Math. Soc. , 20 (1945), pp. 66-67. [10] K. K. FROLOV , Upper estimates of the error of quadrature formulas on classes of functions, Dokl. Akad. Nauk SSSR, 231 ( 1976), pp. 818-821. (In Russian. ) . [1 1] A . O . GEL 'FON D , Differenzenrechnung, Verlag del' Wissenschaften, Berlin,
[7]
1958.
[12] [13]
J. HALTON ,
On the efficiency of certain quasirandom sequences of points in evaluating multidimensional integrals, Number Math. , 27, 2 (1960) , pp. 84-
90 .
H . H A ss E ,
A bstrakte Begriindung del' komplexen Multiplikation und Riemann sche Vermu tung in Funktionkorpern, Abh. Math. Sem. Univ. Hamburg, 10 (1934), pp. 325-348 . [14] E . HLAWI{ A , Zur angeniiherten Berechnung mehrfacher Integrale, Monatsh. Math. , 66 (1962), pp. 140-151 . [15] E . HLAWI{ A , F . FIRN EIS , A N D P . ZINTERHOF , Zahlentheoretische Methoden in del' numerischen Mathematik, Wien-Miinchen-Oldenbourg, 1981 .
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