Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
626 NumberTheoryDay Proceedings of the Conference Held at...
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
626 NumberTheoryDay Proceedings of the Conference Held at Rockefeller University, New York 1976
Edited by M. B. Nathanson
Springer-Verlag Berlin Heidelberg NewYork 1977
Editor Melvyn B. Nathanson Department of Mathematics Southern Illinois University Carbondale, IL 62901/USA
AM S Subject Classification s (1970): 10 D 15,10 E 20,10 L 05,10 L 10,12 A 70
ISBN 3-540-08529-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08529-7 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2140/3140-543210
On 4 M a r c h 1976 t h e l ~ o c k e f e l l e r U n i v e r s i t y h o s t e d a o n e - d a y c o n f e r e n c e on number theory.
The lectures
were as follows:
S. C h o w l a , " L - s e r i e s
and elliptic
curves;" P. ErdSs, "Combinatorial p r o b l e m s in n u m b e r theory;" P. X. Gallagher, " P r i m e s and zeros in small intervals;" C. J. M o r e n o , "Explicit formulas in the theory of automorphic forms;" M .
B. Nathanson, "Oscillations of additive bases;"
and A. Selberg, "l~emarks on multiplicative functions. " T h e field of n u m b e r theory w a s thus fairly broadly represented.
T h e papers in the present v o l u m e are accounts,
several in expanded versions, of m o s t of these lectures.
M.
B. Nathanson, w h o w a s
the original instigator of this n u m b e r theory day, has kindly offered to serve as editor.
W e take this opportunity to m a k e
record of our gratitude to our distinguished
speakers for their participation.
M. S c h r e i b e r 1 S e p t e m b e r 1977
TABLE
OF CONTENTS
i. S. Chowla,
L-series and elliptic curves
I
Z.
P. ErdBs,
P r o b l e m s and results on combinatorial n u m b e r
3.
C. J. M o r e n o ,
4.
M.
5.
A. Selberg, R e m a r k s
theory III
Explicit formulas in the theory of automorphic f o r m s
B. Nathanson,
Oscillations of bases in n u m b e r on multiplicative functions
theory and combinatorics
43 73 217 232
oC-series and elliptic c u r v e s S. C h o w l a
Introduction A. Selherg and S. C h o w l a p r o v e d in Crelle I s Journal, 1965, that if there w e r e a tenth i m a g i n a r y quadratic field Q(V/T-~, corresponding
with c l a s s - n u m b e r
i,
then the
~-series
o~s)
= 2~ ( d ) n - s[ ( _d 1 i s t h e K r o n e c h e r n n
symbol]
1
would be negative at Subsequently, problem
1 s = "~ ,
contradicting
and independently,
A.
( a l r e a d y s t a t e d by G a u s s ) .
quadratic
the "extended Riemann hypothesis".
B a k e r a n d H. M. S t a r k s o l v e d a n o u t s t a n d i n g Namely,
fields with class-number
1.
there are exactly
9 imaginary
G a u s s u s e d t h e l a n g u a g e of b i n a r y q u a d r a t i c
f o r m s instead of that of quadratic fields. The problem
(still unsolved) of the existence (or, otherwise
of rational
points on a given elliptic curve
y
Z
w a s c o n s i d e r e d by Euler, Poincar~, Swinnerton-Dyer,
in m o s t
of certain H e c k e
~-series
3
=x
+ax+b
Mordell,
remarkable
Nagell and m a n y
conjectures,
others.
Birch and
related the o r d e r of the zero
(formed with"GrBssencharaktere")
associated with
certain elliptic c u r v e s to the " M o r d e l l - W e i l " again, w e see the i m p o r t a n c e of the point
IIecke
o~- series Let
~K(S)
rank of these curves. So, h e r e 1 s = -~ for the values of Dirichlet and
~(s). denote the D e d e k i n d zeta function
(N~) -s 72r where
z,Z runs over all integral ideals of the algebraic n u m b e r
field
K.
It is a n a l m o s t u n s p o k e n conjecture - I believe implicit in the w o r k Selberg a n d C h o w l a m e n t i o n e d a b o v e - that
of
if K
is of degree
Z
(there is a m p l e n u m e r i c a l
Serre (in a letter to the author) m a d e
evidence by l~osser, L o w ,
Purdy).
the surprising discovery that
r~K(~) = o for a certain field Q(~'-,
~/~-).
K
of degree
S =~
This field is a quadratic extension of
A proof, with a different example,
All these e x a m p l e s 1
8.
in the study of the
w a s published by A r m i t a g e .
s h o w the special interest that attaches to the point ~-series
of Dirichlet and Hecke.
A n o t h e r a p p r o a c h to the conjectured non-vanishing
of Dirichlet
~-series
o0
>2 X (n)
s
(s> 0;× #×0 )
1 n on the real line
s > 0 is provided by a paper (also in Crelle v s Journal, v o l u m e
dedicated to H. Hasse) by the author,
M.
J. de L e o n and P. Hartung.
R e c e n t notes by John Friedlander and the author (Acta Arithmetica,
Vol.
Z8, P a r t 4, 1976 and G l a s g o w Math. Journal, Vol. 17, 1976) again s h o w the import1 s = ~ for ~-~-series, (in the next line, X denotes a character
ance of the point (rood k))
o0
~-~s) = >2 x(n)s [X ~XO] 1
Let
d
be a p r i m e
number
h(d)
of the f o r m
of Q(~r'~), for
x
Z
n
+ i. S. C h o w l a
x > Z6, is > i.
conjectured that the class-
In fact, for x = Z6,
h(677) = I This is analogous to the G a u s s conjecture that h(-d) > 1 for all d > 163 is square-free).
Finally,
recent unpublished w o r k
zeta function of cubic fields,
when
K
~K(S)
studied in her P h . D .
seems
is a cubic field. H e r w o r k
of Epsteinrs zeta function
of M .
(where
C o w l e s on the D e d e k i n d
to indicate that
combines
the results of special cases of
thesis (Penn State University 1976) with the estimates l Z(s) at s = ~ m a d e by Selberg and C h o w l a in Crellets
d
Journal, 1965, cited above. T h e rest of this p a p e r is divided into five parts:
P a r t I. S o m e
remarks
on the coefficients oo
x
IT,, (l-x n)Z(l-xHn)2 L
c
of the parity of c
n This w o r k is joint with M .
for the first time.
P a r t Z.
1
the p r o b l e m
in the expansion of
0o :Ec
i
l
Here,
n
xn
n
is solved, as far as w e k n o w , Cowles.
The congruence e
is studied.
Here
0-(n) =
n
~ d.
~ ~(n)(mod 5)
[ (n, ii) = i]
This w o r k is joint with J. Cowles.
din P a r t 3.
l~temarks on D e d e k i n d s u m s . A n e w expression is obtained for the c l a s s - n u m b e r
of i m a g i n a r y quadratic
fields.
This is closely linked to recent w o r k of Hirzebruch.
P a r t 4.
On
Fermatls
last t h e o r e m .
This is a n account of recent w o r k with P. C h o w l a ,
linking the study of
F e r m a t ' s equation x p + yP : z p
with the p r o b l e m
(p_>5)
of "rational" points on
y Z : xp + ~ 1
P a r t 5.
l~ecent unpublished w o r k with D. Goldfeld on relations b e t w e e n Epstein's
zeta functions a n d D e d e k i n d zeta functions.
1.
O n the coefficients
c
in the expansion
n
co
x
co
I[
(1-x n)z(1-x lln)z =2
1
§ O.
Let
p(n)
c xn n
1
be defined (Euler) by o0
1
p(n)x n co
1
The problem following
result
II ( t - x n) l
of the parity
concerning
of
p(n)
the value
of
is still unsolved, c
( r o o d Z).
Let
but we will prove p
be a prime,
the
then
n
a) b)
Z divides
c
if
P p ~ 1,3,4,5,
If
p ~ Z, 6 , 7 , 8 , 1 0
( m o d 11) Z
9 ( r o o d 11),
then
Z divides
c
iff
p = u
+ llv Z
(u,v~Z).
P An announcement Ample
support
in the classical the Legendre
Corollary:
of this
paper symbol.
result
has
recently
for this result of Shimura
appeared
is provided
[ 3] .
[ 1] .
in Trotter's
In the following
We give two proofs
table for
(~),
of the foilowing
(p l The
desired
formula
is
nc
n
= c
n
- Z
E
c
u+v=n
n
~(v)
- ZZ
u, v>l
This formula f
and
g
to m e a n
l e a d s u s to i n t r o d u c e
Y u+v=n u, v>l
f(u)g(v).
w i t h itself will b e t a k e n up.
E
u+l]v=n
c
u
~(v).
u, v_>l
the t e r m
"convolution"
of t w o functions
In S e c t i o n 3, the c o n v o l u t i o n of the f u n c t i o n
0-
12 § 2.
T h e proof of (B) From
(A)).
Shimura's paper [ Z] ,
0o
(C)
(assuming
e
X
e
ms = (1-11-s)
m=l
m
.II (1- ~ + -'~gs )-1 p~ll
p
p Z
= (l+ll-S+ll-ZS+ll-3S+...)
where
x
= c p P
P
-s
- p
[-Zs
From
this,
3
II ( l + x + x + x + . . . ) P P P p~ll for
p # 11,
n
c
= p
Zn
c
are inctuded
-
gn+l
r
P
reZn-Zr p
n
= p
Both cases
>2 ( 1 ) r ( z n - r ) r=0
23 (-l)r(Zn+l-r)prcZn+l-Zr r p
r=0
in
rl
[7] (D)
c n=
p
( 1 r n-r rcn-Zr - ) ( r )p P
E
r=0
.
n
The key step in the proof
of
c n
P
-= E p r ( r o o d 5), r=0
for prime
p ~ 11,
is
the foliowing n
E (_l)r(n;r)pr(p+l)n-Zr r=0
Lemma.
Proof: integers
Proceed k < n.
by induction Consider
on
=
n-
the ease
E pr r=0 assume
when
the lemma
n = Zj,
holds for all non-negative
the case when
n
is odd is
simila r.
E r=0
J
E
r=0
(_l)r(n;r)pr(p+l)n-Zr
j-1
( - 1 ) r ( Z J r r ) p r ( p + l ) ZJ- Z r = (- I)0 (~j)p0 (p+l) Zj +
E
r=l
.
.
(-l)r(Z3;r)pr(p+l) Z3-zr
73
(-l)J(ZJ.-J)pJ(p+l)Zj-Zj = (_l)O(Zj-l)pO(p+l) 2j 3 j-I + Z (-l)r[(ZJ-r-l)+(ZJrr?l)]pr(p+l)ZJ-Zr + (_l)J(Zj-~-l)pj (p+l) gj-zj
+
r=l j-1
.
.
j
.
= N (_l)r(ZJ-rr-1)pr(p+l)ZJ-gr + >2 (_l)r(gj~.r/1)pr(p+i)ZJ-Zr r=0
r=l j-i
(_ 1)r((ZJ-1)- r)pr(p+l) (2J -1)- Zr
7"
= (p+i)
r
r=0
n-i
j-1 [--2-1 + (-p) 7" (-1)t((ZJ[Z)-t)pt(p+t) (Zj-Z)-Zt = (p+l) N (-1)r((n-1)-r)pr(p+l) ( n - 1 ) - z r t=0
L
r=0
r
n-2 [--2-] + (-p)
n-i
(_l)t((n- Zt)- t)pt(p+l )(n-g)-Zt
Z
= (p+l)
~
t=0
n- z p
r
- p
r=0
From
(A) and
(D),
it follows that c
~
n
pt
t=0
=
Z
pr
r=0
_=~(pn) ( m o d 5),
for p r i m e
p ~ Ii.
n
F r o m the c a s e
n
m =p ,
j u s t p r o v e d , it is i m m e d i a t e that c
since the
c's
n
- ~(n)(mod 5),
for (n, ll) = 1 ,
are multiplicative and so is ~.
n
Finally note that f r o m f o r m u l a
(C),
it follows that c
not difficult to see that
= i. T h u s it is lln
c n
_ ~( n )(rood 5) ii~
and (=+l)c
- ~ ( n ) ( m o d 5) n
where
ll~In but ii~+I ~ n. From
the first of the two c o n g r u e n c e s above, together with the recursion
formula in the previous section, w e obtain the
Theorem.
~.
5" ~(-~--)(r(v) + 7 ~( u+v=n u u+llv=n ii u, v>l u, v__>l o~.+1
w h e r e Ii ill but ii z
~ i.
u ii
u
)~(v) - Z(n-l)¢(--2---) ( m o d 5) , n II
14
§ 3.
The
c o n v o l u t i o n of
(r w i t h its elf.
In this section w e RamanujanVs
evaluate the c o n v o l u t i o n of
Collected Papers
where
1 o-(0) = ~ ~(-1).
~(u)cY(v) = i.(4)
zeta-function
= (z=)s~(l-s)
s = Z)
-2~(2)
1
~(-1) = - -
above,
1 ~(o)
-
-
g4"
2;
~
= - Zw g
4~ 2
from
From
cr3(n) + ~(0)n~(n)
N o w t h e f u n c t i o n a l e q u a t i o n of t h e R i e m a n n Sir
Hence
5).
~Z(z) " ~(4)
Zr(s)~(s)cos 7 gives (setting
(mod
([I], p. 139), w e h a v e
rZ(z) 2] u+v=n u, v > 0
0- w i t h itself
6
Thus the formula
~(u)~(v) =
-
Z
1 lg
"
of l ~ a m a n u j a n g i v e s
z~(0)~(n) + ~1 (4 )Z -90 '~-~3(n)
i - ~ n~(n)
.
u+v=n
u, v > l
*(u)*(v)
Thus u+v=n
=
1 *(n) + ~15 - ~3(n) - ~n~(n).
57-
Hence,
t a k i n g t h i s e q u a t i o n (rood 5),
u, v>l
Z
o-(u)o-(v) ~ 3o-(n) - 3no-(n) (rood 5) ;
u+v=n
u, v>l
2 u+v=n
u, v > l
e ( u ) ~ ( v ) ~ Z(n-1)~(n) (rood 5) .
15
I~eferences 1.
S. l ~ a m a n u j a n , " C o l l e c t e d
2.
G. S h i m u r a ,
P.
V. S. A i y a r ,
Math. 3.
Papers
of S r i n i v a s a
a n d B. M. W i l s o n ,
A reciprocity
IL~manujan".
Chelsea,
law in non-solvable
New York,
extensions,
I~d. b y G. H. H a r d y , New York,
221(1966), 209-220.
J. Tare, T h e arithmetic of elliptic curves, Inv. Math.
196Z.
J. I~eine A n g e w .
23(1974), 179-206.
16
3.
§ I.
Let
h(d)
On Dedekind
d e n o t e the c l a s s - n u m b e r
sums
of the q u a d r a t i c field
Q(~/d)
and write
k
E ~k ~ (~k) =t
s(h,k) =
{ where I %b(x) = x - [x] - -~ if x ~b(x) = 0 for the D e d e k i n d Further s u c h that
if x
is a n i n t e g e r }
sum. let
t,u
x Z - d y Z = I,
b e the s m a l l e s t positive integral v a l u e s of x , y Here
d
is a positive n o n - s q u a r e
I bz_
/ d = b 0 - bll_
where
the
bls
the " u p p e r "
a r e integers
> Z
and
c o n t i n u e d fraction for
s
simple
....
integer.
1 bs_l -
respectively
Write
1 bs
is the length of the (smaLlest) p e r i o d in
~/d. A l s o let
v/d = a0 + i al+ b e the o r d i n a r y
is not a n integer,
1 az+
c o n t i n u e d fraction for
1 " ' " +--at d
w i t h p e r i o d length
t.
Write
B = bl + b z + --. + b s
A
= a t - at_ 1 + - ... + a I .
We s h a l l s k e t c h the p r o o f s of T h e o r e m s
l a n d 2, f r o m w h i c h T h e o r e m 3
is a n i m m e -
diate consequence. Theorem
i. If d
is a p r i m e
-3(4)
then
3 - g-it + igs(t,u) = 3s - B . u
]Example:
d = 7.
R.S. of (I) = 3. Z -
Here
t = 8, u = 3;
1 1 ~/7 = 3 - ~-- ~-
(3+6) = - 3
L.S. of ( 1 ) - - 3 - 5 - - + - 5 -
{ *( ) + Z~(
)} = - 3 .
(l)
17
Theorem
Z.
If d
is a p r i m e
-3(4),
then
(z)
3s - B = - A
(many when
d
examples,
is a p r i m e
this a n d c o m b i n i n g
Theorem
3.
including
~ 3(4)
and
theorems
If d
d = 1019, w e r e
h(d) = i,
c h e c k e d by P.
Hirzebruch
Chowla)
p r o v e d that
3h(-d) = A.
Using
1 a n d Z w e obtain
is a p r i m e
~'3(4)
then if h(d) = l
3 - Z_jt + IZs(t,u) = -3h(-d)
(3)
u
Examples:
This applies to all p r i m e s
O n e can check,
§ Z.
We
as a n e x a m p l e ,
d ---3(4)
the case
Dedekind
"Analytic N u m b e r
Theory"
1955), p. Z56.
Let
H-function.
1 a n d Z.
except
d = 79.
T h e y a r e b a s e d on the theory
Recall the following f r o m
(Tara Institute of F u n d a m e n t a l
a,b,c,d
I00,
d = 19.
shall sketch the proofs of T h e o r e m s
of the w e l l - k n o w n
less than
be positive integers with
l~ademacher's
Research,
ad - bc = i.
Bombay,
1954-
T h e n with
lm(~-) > 0
,aT+d, log ~tc---~--~j = log ~(T) 1
c~+d
(4) wi
+~log F-- +~(a+d) -
wi s(d, c)
(there will be no d a n g e r of confusing the
d
h e r e with o u r previous
d).
O n the
other h a n d s u p p o s e
aT+b cT+d - b0
Then
I b~
i b Z-
I "'" - (bs+T)
one easily a r g u e s that (aT+b) 1 log ~3 ~ - log 13(7) - ~ log(cr+d)
wi
--
wi
4 s +~-
(bo+bl+..°+bs)
(5)
18
§3.
We apply
so that
(4)
and
(5)
with (here
we write
d = N)
a =t,
b=Nu
c=u,
d=t
a d - b c = 1. Let (wi~integral
b ' s ~ Z)
Nu T
l = b0 - b 1-
1
1
b Z-
bs
(bo=b s) •
T h e n (the bracketed portion is the "period") 1
1
~/N = b 0 - bl -
Comparing obtain
§ 4.
(4)
and
(5)
(1) of T h e o r e m
Theorem
It states
ba -
in our special
1
...
case
b
(N
1 Zb
s-l-
s
is a prime
of t h e f o r m
4k+3)
we
1.
g is proved
that (see pages
by using the famous
Reciprocity
Z59 a n d Z57 of 1 K a d e m a c h e r J s
Law for Dedekind
Sums.
book cited above)
1 1 (_d c i / s(d,c/+ s(c,d/ =- ~+TZ- c + ~ + c d when
c,d > 0
and
(c,d)
law allows us to calcuiate Also
s(-c,d)
can be built.
= -s(e,d).
= 1.
Since
s(c,d)
rapidly the values These
are
has a period of
s(c,d)
the main ideas
c
in
when
c
d, and
the reciprocity d
are
large.
o n w h i c h a p r o o f of T h e o r e m
3
19
Z 4.
§ I.
Write
(p
The
non-trivial
is a n o d d p r i m e
rational points in y
= 4x p + 1
> 3)
x p + yP = l
(F)
and y
We and
(H).
on
are
o n the c u r v e with
x = i,
on
(H)
main
y = 0
a non-trivial " Q - p o i n t " ] converse
§ Z.
We first
Theorem
i.
.
and
x = O,
now
x = 0,
on a "Q-point" on
(F)
but p e r h a p s
we
mean
or "trivial" Q - p o i n t s
y = I.
y = -1.
on
(F)
or
implies one on
(H)
(H)
will refer to
and conversely.
prove
F => H. on
F
implies
one on
H.
From = 1
on squaring (xP-yP) x + 4(xy) p = 1
Set x p - yP
= u
x y = -V. So
"obvious"
Q)
(F)
not entirely trivial.
x p + yP
follows,
The
y = 1 and
A Q-point
is e l e m e n t a r y ,
T h a t is a Q - p o i n t
Proof.
y ~ Q.
(with coefficients in
in
are
result is that a [ f r o m
The
f(x,y) = 0
x c Q,
x = 0,
Our
(H)
of p o s s i b l e non-trivial " Q - p o i n t s "
B y a Q - p o i n t o n the " c u r v e " (x, y)
Those
= 4x p + 1
a r e interested in the p r o b l e m
a point (F)
Z
(i) b e c o m e s
(1)
20
u So if x,y ~ Q
in (i), then
u,v { Q
2
(z)
= 4v p + 1 .
in ( 2 ) .
q.e.d.
So, the first half of our assertion at the end of 1 is trivial.
We n o w prove:
~3. Theorem
Z.
H~>
F.
T o this end, w e set in
(Z), ce
u
-f
2'
(3)
v=8
with
~,p,,~, 8 { z;
So
( ~ , P ) = ('l, 8) = 1
(4)
(Z) b e c o m e s Z Z = 4(~6 )P + 1
(5)
P or
~26P
Since the r.h.s,
= 4~Zy p + ~ Z 6 P .
of (i) is - 0 ( ~ z) w e obtain
~Zl~26P.
(6) But
(ce,~) = 1 and so,
~218P. O n the other hand f r o m
(6), 6P(~2-~ Z) = 4~Z'f p.
8P]4p z
§ 4. Case
We
n o w distinguish
A:
6,
in
So, since
(~/,6) = 1 (8)
Z cases:
(6), is odd
In this case it follows f r o m
(8) tha t
6Pip z From
(7)
(7) and
(9)
{9} w e obtain 2 = 8p .
(10)
21
So Z =~ ,
with
5 = ~i
(n)
Pl ~ Z. By (ill
(6)
becomes = 4y p + p
(iZ)
Or
(13)
(~-~P)(~+~5 p) = 4y p .
Now odd.
~
From
is odd from
(1) a n d
(10)
since
(1Z) ~1 i s o d d .
8
is odd.
Since
~
and
So from ~1 a r e
(11) a n d odd,
(13),
it follows
a
is
from
(13) that
- ~1p = z'~p'
From
~ + ~1p = z~p I ~ i Y z ' z , ,q,t z = 81
(i4)
(14)
Z~ p = Z(,~Z-yi P P)
(i5)
I. e.
P P P ~31 + Y1 = YZ i.e.
(16)
(F) has the Q-point
~I x=
1
Z'
Y-
Y Thus, in case
Z Y
(A)
(H) --~ F .
§5.
Case We
(B): 8,
in
(6),
is e v e n .
shall use the notation
q~llM to m e a n
that, with
q a prime,
we have
22
qlM i.e. q
of
is the highest In this case,
power namely
of
q
B,
c+llM
but
dividing
M.
In our application,
q=Z.
let
zelrs
(17}
i. e.
5 = zcs1
Then, f r o m
(18)
(5 1 o d d ) .
(6) 413Zy p _- 5P{of2_[32).
(19)
Then is odd 5'
Since
in
(19) h a s
Zp
[since
as a factor
and
(zo)
(~, 6) = 1] .
p > Z it follows
from
(19)
or
(ZO)
that [3 i s e v e n
[so Of is odd, since
(of,[3) = i]
(18) w e n o w h a v e
From
zcP-elf[32 I.e.
(zl)
(21)
s erring c=Zd.
(z2)
Z d p - l ] [ [3 .
(z3)
We have
So,
set [3 = 2dP-l[31
5 = 22dst
the latter from So
(18)
and
([31 odd),
(8 1 odd),
(ZZ).
(19) gives (cancelling out 22dp
f r o m both sides)
(24)
(25)
23
=
z
pl)
(26)
i. e. Z6p Z 1 =~ Yp+ Since
r.s.
zZdp-Z Z p ~161
(Z7)
of (Z7) is ~ 0(~iZ) w e get since ~ is p r i m e to ~, Z
and so to ~i" that
p
(28)
~iTbl • Also, f r o m
From
From
(Z6) since
(Z8) and
(26) and
6 is prime to ~/ (and so
6 1 is prime to "~) that
2
(29)
p Z 61 = ~I"
(30)
(Z9),
(30),
u p = (=z-zZdP-Z~Z).
(31)
So (~+ Z dp- i~i )(~ - Z dp- I~i) = U p . Since
6 was even (hypothesis of case B, first line of §5), "~ is odd [since
I] so each factor on the ~. s. of (3Z) is odd using are relatively p r i m e since Thus, f r o m
(=,~)
=
(3Z) ('~,6 ) =
(21) above; also the Z factors
i.
(3Z) + zZdp-I~I = YIP
(331)
+ zZdp-I~I = -~P.
(33 Z)
[ %ve used, in (3Z), that if the product of Z relatively p r i m e nos. is a pth-power, then each no. is a p-th power] . Subtracting
(33Z) f r o m
(331) w e get zZdP~I
(34)
24
But ~i
is a p-th p o w e r
from
(30),
~l: ~pl From
(34)
and
say
[~l~ z]
(35); P P = (zZdNI)P . Y2 - Yl
i° e.
(36)
(F) h a s the " Q - s o l u t i o n "
YI
YZ X
-
Zk I
Thus
(35)
H
=> F
(above proof).
Since
Y -
F => H
F
Corollary:
Fermatls
Last Theorem
(non-trivial)
(§Z)
it follows that
H.
is true if a n d only if the c u r v e
y
h a s no rational points
ZX I
Z
= 4x p + l
on it.
(37)
25
5.
O n the twisting of Epstein zeta-functions into Hecke-Artin
§ I.
L - s e r i e s of K u m m e r
fields
T h e Epstein zeta-function
Z(s, C) = ~' (ax 2+ bxy+ cy 2) s
w h e r e the s u m m a t i o n
is o v e r all integers
x,y
excluding
x = y = 0,
with associated
binary quadratic f o r m Z
C = ax
Z
+ bxy + cy
a n d dis criminant
Z~ = b Z - 4ac < 0 is so defined for
l~e(s) > i,
and by analytic continuation over the w h o l e s-plane.
H e r e , by a b u s e of language, Z 3 + bxy + cy .
C
refers to the set of integers representable by
ax
We integer).
a r e c o n c e r n e d with
Z(s, C)
in the special case w h e n
A
= -3k Z
(k,
an
In this case, G a u s s and D e d e k i n d noted the connections b e t w e e n these
functions a n d the law of cubic reciprocity.
More
explicitly, D e d e k i n d p r o v e d that
(Crelle's Journal, 1900)
Z~K(s) ~(s)
where forms
=
i
~I
.
_
1
~I
(xZ+ZTyZ)S
(l)
(4xZ+ Zxy+TyZ)S
3 ~i4(s) is the D e d e k i n d zeta-function for the field K = Q(~/2). Note that both 2Z Z 2 x + Z7y and 4 x + 2xy + 7y h a v e discriminant -108 = -3.63 . This rela-
tion implies that for p r i m e s
p ---1 ( n o d 3),
Z is a cubic residue
( m o d p) if and
only if
p = x This r e m a r k a b l e
result is due to Gauss.
Z
Z
+ Z7y . We
quote f r o m D e d e k i n d ' s p a p e r (pp. Z06-
207 of his Collected Papers). O b s ervatio venustis s i m a inductione facta Z
es__~tl~esiduurn vel n o n R e s i d u u m formae
3n + I,
prout
p
cubicum numeri primi
representablis est p e r f o r m a n
p
26
x x + Z7yy
vel
4 x x + Zxy + 7yy. 3
per
est R e s i d u u m xx + Z43yy
vel n o n R e s i d u u m ,
au___~t4 x x + Zxy + 61yy
prout
p
representabilis es__~t
ve__! 7xx + 6xy + 36yy aut
9xx + 6xy + Z8yy. (Note that the f o r m
§ Z.
Let
S
13x Z + 4 x y + 19y Z of discriminant
-972
h a s b e e n omitted~ )
be a set of integers such that
S = Cl[.J CZ~_) C 5 ... ~J C H
where
each
C. is also a set of integers and J C 1.... , C H
f o r m a multiplicative g r o u p We
G.
define co D(s, C.) = E J n:l
w h e r e the
c.(n) ,I ns
cj(n) are arbitrary c o m p l e x n u m b e r s ,
the Dirichlet series associated to co
C.. 3
Let
a
H
and
cj(n) = 0
if n ~ C.,3 to be
n = Z~ D(s, C . ) . n=l n s j =i J
B y a "twist" of the left side of (Z), w e m e a n
(Z)
the n e w series
H
x (C.)D(s, C.) j=l
~
J a
where
X
is any non-trivial character of G.
In general, if >2.--Sn has a n E u l e r n S
product, then the "twisted" series also has a n E u l e r product. A s a n e x a m p l e of a twist let S of the following
3 forms:
be the set of integers representable by a n y
27
C I = x Z + Z7y Z
C Z = 4 x Z + Zxy + 7y 2
C 3 = 4 x Z - Zxy + 7y Z
s o that
s = ciU czU ¢3"
(3)
T h e associated Dirichlet series In this e x a m p l e ,
(E'
D(s,C.) are nothing but ]Epstein zeta functions. J the analogue of (Z) is
i )(I-3-s+3. 9 -s) = Z ( s , C I) + Z(s, C Z) + Z(s, C3). (xZ+3yZ) s
(4)
This is p r o v e d as follows: In the s u m i
I~ = E'
(5)
(xZ+3yZ) s
the variables into 4
x,y
ranging over all integers excluding
classes, so that R=A+B+C+D
where
A =
I
~'
31x
(xZ+3yz)s
Sly B=
1~' 3Ix
i (xZ+3yZ) s
3 ,y
31y = ~(3-s_9 -s) ,
-R"
9
- s
x = y = 0,
can be divided
28
C :
i
~'
3~'x (xZ+3yZ)s ?[y
:
~' -
31y
2'
31~ 3[y
= Z ( s , Cl) - 9-s1%, D =
[
2;' 3~'y (xZ+3y2) s 3~y
=Z
---1(3) /x >2v +y--2(3) E' ] \~---I(3) y =-z(3)/
=2 ~y(3) Now,
3 Ix
3ty
in the s u m i x=-y(3 ) (xZ+3yZ) s
w e just m a k e
the transformation X = U +
ZV t
y
:U-
V
w h i c h leads to D = g(Z(s, Cz)-A). Hence
(4) is proved,
by using the relation
1% : A + B + C + D.
Let
x (c l) = t x ~ I+ VZ~X (C z) = x ( C 3) :
J
be a character of the composition group quadratic f o r m s
of discriminant
-108,
Z
{CI, Cz, C 3 ]
of reduced primitive binary
satisfying the following relations
29
Z
C1
= CI
Z
C Z
: C3
C3 Z = C 2
CzC 3 : C 1 •
T h e structure of the a b o v e g r o u p is d e t e r m i n e d by the fact that if n, is in C. and 1 1 in C. then n.n. is in C.C.. 3 J ij 1 3 In Section 8 of this p a p e r it will be p r o v e d by a novel m e t h o d that the "twist"
n
of (~,
1 ) (i_3-s+3. 9-s) (xZ+3yZ) s
is nothing but
%K(S) This, of course,
gives D e d e k i n d ' s result
(i), and the m e t h o d
extends to m a n y
other cases.
§ 3.
W e w o u l d like to ask the follo~ving question.
E u l e r product) on
~K(S)
where
K = Q(%/~,
zeta functions of sub-fields of K u m m e r
Is any "twist" (which has an
a ratio of products of Dedekind's A positive a n s w e r
fields?
is supported
by the e x a m p l e s that follow. A Kurn~er
where
k
answer,
§4.
and
a
field is
are positive integers.
If the a b o v e question does h a v e a positive
then it is likely that 3 Ik.
This section is devoted to s o m e
tive binary quadratic f o r m s
special e x a m p l e s .
of discriminants
-Z700
sition g r o u p tables, respectively. Discriminant
-Z700
C 1 = (i, 0,675) C z = (Z5, 0, Z7)
and
We
list the r e d u c e d p r i m i -
-18ZbZ,
and their c o m p o -
30
C 3 = (13, 2, 52) C 4 = (4, Z, 169) C 5 = (7, 4, 97) C 6 = (9,6,76) C 7 = (19, 6, 36) C 8 = (25, i0, 28) C 9 = (ZS, Z0, 3t)
CI0 = (27, 18, ZS) and, for
3 < n < i0,
Cn+8 = ~n" Here,
if
C = (a,b,c),
then
= (a, - b , c). { T h e notation,
(a,b,c)
for the binary quadratic f o r m
ax
2
Z + bxy + cy , is a
standard one. } G r o u p Table for D i s c r i m i n a n t T h e generators are
C3
C I = C30
and
-2700 C7 .
C 6 = C72C32
3 C Z = C3
C7 = C7
C3 = C3
2 C8 = C7 C3 2
Z
C4 = C3
C 9 = C7C 3
C 5 = C72C33 a n d for any
n
CI0 = C 7 C 3
(l 0 by a quadratic f o r m in an even n u m b e r of variables > 4 is a linear combination of multip[icative arithmetic functions. A s an example, if w e combine equations
(i) and
(4), and use the following
relation (here I< = Q(~/Z))
1 ~K(s) = ~(s)
[
z 1] I] II x ( m o d 3) w=l p
( i-
x(p)× P' pS
3(z)
-i (7)
co :
m
O(n)
n=l n
s
which has been derived in Section 6, w e obtain
i~
z
xZ+z7yz(n) = ~
m (-i08)
E
kl n
I l~4xZ+Zxy+TyZ(n) =-3Z
where From
+ ZO(n
(8)
k
-I08 )] kln~ (--~) - 0(n
i~ z(n) is the n u m b e r of ways n is representable by a x Z + b x y + cy Z. axZ+bxy+cy (7), it can easily be shown that 0(n) is a multiplicative function and defined
on the p r i m e powers
p
as follows:
39
-l,
f
0(p ~ ) =
where
(3 ~)
derive
§ 8.
is the Legendre
many
equality between
finite number relevant stated
I
--" Z(3)
1,
~ --" 0(3)
In a s i m i l a r (8)
series
(e. g. p r i m e s
it is also possible
of discriminant
fields).
with Euler
Z.
In t h i s p a r t
products
may
argument
say.
of the
exclude a
that divide the discriminant
With a special
to
- 3 k z,
of certain
these factors
can be
exactly. With this convention,
(4)
gives oo
3
Z~(s) 2 (-~)~ 1__= 1
using a well-known Euier
fashion,
for forms
oroof at the end of Section
two Dirichlet
number
if x3 ~ Z(p)
I+(~)+... +(~)~ if x 3--z(p)
of the type
of Ubadn primes
algebraic
0,
symbol.
other formulae
We now give the promised
paper,
~ =-1(3)
product
theorem
n
of D i r i c h l e t .
for the left side of
(9)
n
s
~ Z(s,C.), j=l
(9)
J
Here
(~---) i s t h e K r o n e c k e r n i s up to l o c a i f a c t o r s (p=Z, 3)
symbol.
The
3
Z
II
( 1 - p - S ) - Z II ( 1 - p - Z S ) -1 = E Z ( s , C.) .
p---z(3) For
the s a k e of clarity w e
j=l
p
(lO)
J
r e p e a t that Z C I = x
2 + Z7y
C Z = 4 x Z + Z x y + 7y Z C 3 = 4x 2 - 2xy + 7y Z
a n d that t h e s e
G's
earlier a c h a r a c t e r
form
a multiplicative g r o u p ,
o n this g r o u p
as r e c o g n i z e d
is defined as follows:
× ( c 1) = 1
× (C Z) = Z × ( C 3) = ~
.
by Gauss.
As noted
40
_i+v4"y Here
~
=
- -
•
13 p~l(3)
a c u b e root of unity.
x(p)/_ z
1-
pS
]
p~ C 1
13 p-t(3)
We
now
(i - ll,3,n pp
( 1 )_l 1 - --~s p
(n)
pc C 3
pc C 2
13 p---Z(3)
introduct the "twist"~
3
= ~ x(Cj)Z(s,c.). j=l
3
It is clear that
X(P) : l if p c G 1 X(P) = ~
if p c C Z Z
X (P) = ~
We
observe
is that if p
if p c C 3
that the "twist" d o e s not t o u c h p r i m e s is a p r i m e
The
r e a s o n for this
--- 2(3),
tions b y the totality of
G
representations
3 C.;
b y the
p =- 2(3).
a n d if the positive integer n h a s r(n) r e p r e s e n t a 2~ (j=l, 2,3), then n . p h a s the s a m e n u m b e r , r(n), of and, m o r e o v e r ,
there is a w e l l - k n o w n
I-i
correspond-
e n c e leading to
r(n.p Since
C 2
o n the left side of
and
C 3
) = r(n),
if p --- 2(mod 3).
a r e "identical"
(ii) c a n b e c o m b i n e d ,
13 p-l(3)
1 -
1
13 p~l(3)
pc C 1
(in a s e n s e ~ )
1 -
a~
3 Y
j=l
We
recall equation (7):
×(c.)z(s,
3
a n d third factors
to give
pc C 2
=
the s e c o n d
c.). 3
1 - pS
13 1 p---Z(3)
(12)
41
~(s)
where
K = Q(3X//Z),
[
II
II
YI
× mod 3
w=l
p
(
1-
X(P)Xp'3(Z) ) p S
and
if x 3 ---Z(p) soluble
×p, 3(z) =
~izwi e
3 if x 3 ~ 2(p) .
Comparing this with (12) w e get
~K(S)
3
z x(Cj)Z(s,C)j -- 2--~(s) j=l where
K = Q(~/2).
Equation
(i), up to local factors, follows, on noting that x(Cz)Z(s, C 2) + x(C3)Z(s,C 3) = -Z(s, CZ).
§ 9. For the discriminant -1825Z, w e omitted in Section 4 to give the "twist" for the case Q(~/Z6).
Z(s,C I)+ Z(s, Cz) - Z(s,C 3) - Z(s,C 4) - Z(s,C 5) + ZZ(s, C6) - Z(s, C7) + ZZ(S, Cs) - Z(s, C9) + ZZ(s, CI0) - Z(s, CII ) + ZZ(s, CIZ) - Z(s, C13) + ZZ(s, C14) - Z(s, CI5) - Z(s, CI6) - Z(s, CI7) - Z(s, CI8) - Z(s, CI9)
~K(S) with
31
K = W( 3) w e found
In the case
H o w m a n y linear A
(i. e. , with the
A = -108 pZ (p, a
42
1 + h(-a) Z
such combinations.
T h e s e appear, to us, to be the only ones.
Finally, it is fitting to mention that the importance of the concept of twisting w a s first recognized by A. Weil in his paper "Bestirnmung der Dirichletschen Reihen durch ihre Funktional-Gleichungen"
(Math. Annalen,
1969).
Problems
and results on combinatorial n u m b e r
theory III
Paul ErdSs
Like the two previous p a p e r s of the s a m e II) I will discuss p r o b l e m s
in n u m b e r
title (I will refer to t h e m as I a n d
theory w h i c h h a v e a combinatorial flavor.
T o avoid repetitions a n d to shorten the p a p e r as m u c h previous results w h e n e v e r
convenient and will state as m a n y
possible, and will discuss the old p r o b l e m s some
as possible I will refer to new problems
as
only w h e n they w e r e neglected or if
n e w result has b e e n obtained. P. E r d ~ s ,
Problems
and results on combinatorial n u m b e r
theory I and II,
a s u r v e y of combinatorial theory, 1973, N o r t h Holland, I17-138; J o u r n 4 e s Arithm & t i q u e s de B o r d e a u x papers have many hombres,
Juin 1974, A s t 4 r i s q u e Nos.
references.
Monographies
(1963), 81-135.
Graham
Z4-Z5,
Z95-310.
See also Q u e l q u e s p r o b l ~ m e s
de i' E n s e i g n e m e n t
Math&matique
Both of these
de la th4orie des
iNo. 6, Univ. de G e n e v a
and I will soon publish a p a p e r w h i c h brings this p a p e r up
to date. P. E r d B s , S o m e a n d Publ. Math. I.
unsolved p r o b l e m s ,
Inst. H u n g a r .
Acad.
M i c h i g a n Math.
J. 4(1957), Z91-300
Sci. 6(1961), ZZI-Z54.
First I discuss V a n der W a e r d e n ' s and S z e m e r ~ d i ' s t h e o r e m
tions.
D e n o t e by
exceeding
and related ques-
f(n) the smallest integer so that if w e divide the integers not
n into two classes then as least one of t h e m
p r o g r e s s i o n of n t e r m s .
More
generally, denote by
contains an arithmetic f (n) the largest integer so u
that w e can divide the integers not exceeding every arithmetic p r o g r e s s i o n of n
f (n) into two classes so that in u n+u e a c h class has f e w e r than --7- terms.
terms
T h e best l o w e r b o u n d for f(n) is due to B e r l e k a m p , (f(p) > pZP
if p
is a p r i m e a n d
to decide if f(n)I/n -- o0 is true.
f(n) > cZ n My
for all n).
L o v ~ s z and myself,
It w o u l d be v e r y interesting
g u e s s w o u l d be that it is true.
I p r o v e d by the
probabilistic m e t h o d that fu (n) > (l+E c )n if u > cn. T h e proof gives nothing if u / is 0(nl/Z). It w o u l d be v e r y interesting to give s o m e usable u p p e r a n d l o w e r b o u n d s for
f (n). A s far as I k n o w
(Bull. Canad.u Math. fz(n)
the only result is due to J. S p e n c e r w h o p r o v e d
Soc. 16(1973), 464)
fl(n) = n(n-l),
equality only if n = zt"
is not k n o w n . F o r various other generalizations (see II). D e n o t e by
rk(n) the smallest integer so that every s e q u e n c e
1
___I
n
n.
nl=n
the m i n i m u m
gives a
Put
u
where
n1
1
is to be taken o v e r all covering
systems
a . ( m o d n.). 1
I conjecture
1
that
(Z)
u
n
-oo
as
n-oo.
If (Z) is true it w o u l d be interesting to estimate
u
from
a b o v e a n d below.
n
Put
f(n) = m i n k
vchere the m i n i m u m
perhaps
F(n) = rain a k
is extended o v e r all s y s t e m s
interesting to get non-trivial b o u n d s for Here
and
it is w o r t h w h i l e
(I) with
f(n) a n d
n I = n.
It w o u l d be v e r y
F(n).
to introduce a n e w p a r a m e t e r .
u (c) = m i n > n
1 n.
Put
n[ = n
'
1
where
the m i n i m u m
is extended o v e r all finite s y s t e m s
a . ( m o d n.),
n = n I 1 are the divisors of n.
On
1
the other h a n d B e n k o s k i a n d I conjectured that if cr(n)/n > C distinct p r o p e r divisors of n. smallest value of
C
then
n
is the s u m
of
If this conjecture is true w e w a n t to estimate the
for w h i c h the conjecture holds.
A n older conjecture of B e n k o s k i
states:
if n
is odd a n d
0-(n_._._~)> Z then
n
n
is the s u m
of distinct p r o p e r divisors of
n.
O n e can also study infinite covering his students but to avoid trivialities satisfy a c o n g r u e n c e
m
gruences
if k > k0(E)
i
--
done by Selfridge a n d
every
m
> m 0
must
A n o t h e r possibility w o u l d be to
1
the density of the integers satisfying none of the con-
a . ( m o d n.) 1 < i < k I
as w a s
one usually insists:
~ a . ( m o d n.), n~ > n . I
require that
systems
is less than
E .
Perhaps
the first condition implies
I
the second. D e n o t e by P1
N
the s e q u e n c e
I < nl
a n d sufficient
of integers
m
n.. 1
a n d sufficient condition that a l m o s t
all
(4)?
F o r particular choices of the decide if a l m o s t all integers
a. (say a. = 0) it often is v e r y h a r d to i 1 satisfy one of the c o n g r u e n c e s a.(modl ni)" A v e r y
old p r o b l e m
Is it true that a l m o s t all integers h a v e t w o divisors
of m i n e
states:
d I < d z < Zd I. If this conjecture is correct one could c h o o s e as m o d u l i the integers w h i c h are minimal
relative to the property of having t w o divisors
in the s e n s e that no p r o p e r divisor has that property. determine
Many
dI < d2
o log log n 105
~ai}
2k + p = n
Unfortunately to decide questions I p r o v e d that for infinitely
but could not decide w h e t h e r
is the largest integer for w h i c h all the n u m b e r s
f(n) = 0(log n).
primes.
I am
fairly certain that this conjecture is true.
likely that for infinitely m a n y
squarefree.
n
all the integers
I conjectured
n - 2k, 1 < k < log n --
seems
for
i = l,...,k? D e n o t e by
many
a p p a r e n t if w e p o s e
be a n y finite set of p r i m e s .
that e v e r y sufficiently large odd integer is of the f o r m every
2k + @
Is there in fact a n odd integer not of this f o r m ?
T h e connection with covering c o n g r u e n c e s the following question:
(using
p + 2
Is it true that e v e r y sufficiently large o d d integer is of the f o r m @
r
or f e w e r
doubtful if covering c o n g r u e n c e s
In the opposite direction Gallagher
of Linnik) that to every
r
D o they contain an
Schinzel p r o v e d that there a r e infinitely m a n y
density of integers of the f o r m
where
has at
Is it true that for every of a p r i m e
Is the density of these integers positive?
p + Zk + Z £.
the m e t h o d
v e r y difficult:
o d d integers not the s u m
infinite arithmetic p r o g r e s s i o n ? will help here.
r there is an arith-
factors.
there are infinitely m a n y powers
2k + O
systems
log
Z
O n the other h a n d it n - 2k '
Zk < n
are
are
51
Incidentally I a m
sure that lira (ai+1 - a.) = oo.
This would certainly
follow if there are covering s y s t e m s with arbitrarily large
n I.
T h e following s o m e w h a t v a g u e conjecture can be formulated.
Consider all
the arithmetic progressions (of odd n u m b e r s ) no t e r m of w h i c h is of the f o r m 2k + p.
Is it true that all these progressions can be obtained f r o m covering
congruences and that all (perhaps with a finite n u m b e r in any of these progressions are of the f o r m
of exceptions) integers not
Zk + p?
Finally C o h e n and Selfridge proved by covering congruences that there is a n arithmetic progression of odd n u m b e r s
no t e r m of w h i c h is of the f o r m
Zk + p ~
and Schinzel used covering congruences for the study of irreducibility of polynomials.
References P. ErdSs, O n integers of the f o r m Zk + p and s o m e related problems, Summa Brasil Math. 2(1950), 113-123. F o r further literature on covering congruences see P. ErdSs, S o m e p r o b l e m s in n u m b e r theory, C o m p u t e r s in n u m b e r theory, Proc. Atlas Syrup. Oxford 1969 Acad. P r e s s 1971, 405-414. A. Schinzel, Reducibility of polynomials, ibid. 73-75. F. C o h e n and J. L. Selfridge, Not every n u m b e r is the s u m or difference of two p r i m e powers, Math. of C o m p u t a t i o n Z9(1975), 79-8Z.
52
4. An if n o 1 ~--< ai
Some
unconventional
infinite s e q u e n c e
extremal
I_< a I < ...
problems
of i n t e g e r s is called a n
A
sequence
a. is the distinct s u m of o t h e r a's. I 0 r o v e d that for e v e r y A sequence I 1 i00. Sullivan o b t a i n e d a v e r y substantial i m p r o v e m e n t , he proved ~--< ai
It w o u l d
b e interesting to d e t e r m i n e
z!
max
where
the m a x i m u m
4.
is e x t e n d e d
ai o v e r all A
sequences.
greater than
Z.
b I < b Z < ... some
so that t h e r e s h o u l d b e a n
absolute constant
the o t h e r s e q u e n c e s Perhaps A
Sullivan c o n j e c t u r e s that this m a x i m u m
Is it p o s s i b l e to obtain n e c e s s a r y
sequences
c
and every
considered
A
n v
sequence The
the inequalities of L e v i n e
(see their f o r t h c o m i n g
E 1 ai
shows
< log Z + ~
that this is best Usually
and
one
the
is rarely
Here mentioned
Another Let
as
As
I r e f e r to this p r o b l e m
as
(I). (I) for
is a n
t e n d s to infinity,
npn
+
A
sequence
I,...
,
Zn
of these
extremal
and
of i n t e g e r s
Ryavec
problems and
others
s u c h that all the s u m s
is difficult proved n i
that if ~iai '~ i =0
n ~ -!-I < Z - zn_----l 7- equality if a n d only if a. = zi-l. -i i=l ai oldest p r o b l e m s
i _ < a I < ... < a n _ < x
Is it true that far as I k n o w
c o u p l e of s i m p l e i< a I
n
k
for
but I h a v e not b e e n able to p r o v e this.
n > no(k)
By a remark
of V~. Straus it holds for
k=Z. #
It is not h a r d to see that of
F(n)
exp (log n) c ,
(i) holds for e v e r y T h e following m o r e
be a g r a p h of n integer
Perhaps
the true o r d e r of m a g n i t u d e
is
(i)
perhaps
F(n) I/n --1.
c. general p r o b l e m
vertices and
k
x i,x i ~ x., 1 < i < j < n. j
the t w o integers
-1
edges.
j
and
m i g h t be of interest.
T o e a c h v e r t e x of G
If x.
is joined to x.
~
J
_
x. + x.
c > 1
x.x.. 1 3
T h u s w e associate
Let
G(n;k)
w e associate an
w e associate to the edge Zk
integers to the g r a p h
61
G(n;k). ing to and
D e n o t e by G(n;k).
A(G(n;k))
the smallest n u m b e r
if log log kn -- Z
Perhaps
then
of distinct integers c o r r e s p o n d -
A(G(n;k)) > n Z-E
for every
n > nO.
conjecture
This conjecture if true is a far reaching extension of m y Z-c fz(n) > n
All these conjectures
c a n be extended to the c a s e w h e n
the
x.
G > 0 original
a r e real
I
or c o m p l e x
numbers
or e l e m e n t s
For a few weeks
of a vector space.
I thought that the following result m i g h t hold (here
a n d our g r a p h is regular of d e g r e e one). integers. the
Zn
T h e n there a r e at least numbers
Let
I
i
al,...,an;
(or at least
{ a +b ,a.b.}, i = I,Z ..... n. I
too optimistic.
n+l
A.
cn)
b I.... ,b n
llubin s h o w e d t h a t
numbers
amongst
the
among
I was much
I
T h e conjecture certainly fails for
can be real n u m b e r s
Zn
distinct n u m b e r s
c > I/Z
a n d if the !
b's
be
n = Zm
then there d o n o t h a v e
{ai+bi, aibi}.
to be m o r e
It is a l m o s t
than
cn I/Z
certain that the s a m e
a.'s
and
i
distinct holds if
the
a. a n d b. are restricted to be integers, but as far as I k n o w R u b i n did not l i i+~ yet w o r k out the details. If w e a s s u m e k > n or p e r h a p s only k / n --0o one p e r h a p s m i g h t get s o m e
results but I do not h a v e a n y plausible conjecture
so far.
62
Some
8.
unconventional
Is t h e r e a s e q u e n c e
aI < a Z
O. i"i=0o
It is likely that
(3)
lira inf f(n) = 1 , n=o0
Perhaps
f(n) = o(log log n).
inaccessible there
states:
is a
y < x
A weaker
To every
¢ > 0
conjecture there
is an
which x0
is perhaps
so that for
not quite every
x > x0
so that
(4)
~(x) - ~(y)
O.
but this is e v e n h a r d e r
than
prove 1
~(x)
I conjectured
1 ~i p-pj
(6)
where
o n c e optimistically
in
~i Pj < p " log p.
fZ(p) -- 1 . p< x
that
- 1 + o(i)
(6), if true, is of c o u r s e
hopeless.
f(n).
I could not
64
H e n s l e y and P~ichards r e c e n t l y s h o w e d that if the p r i m e k - t u p l e i s t r u e (in f a c t i t c e r t a i n l y " m u s t " there are infinitely m a n y absolute constant
x
for w h i c h
y
~(x+y) > ~(x) + it(y), and in fact for an
c > 0.
(7)
w(x) + w(y) + c Y / ( l o g y) < w(x+y) .
l~ichards a n d I h a v e a f o r t h c o m i n g Monatshefte
der Mathematik.
There
p a p e r on s o m e
of these questions in
is an i m p o r t a n t d i s a g r e e m e n t
l~ichards believes that (7) holds for arbitrarily large values of x
conjecture
of c o u r s e b e t r u e ) t h e n f o r e v e r y l a r g e
and
y.
c
b e t w e e n us. a n d suitable
I conjecture the opposite.
O n e final conjecture: consecutive p r i m e s
in
Let
n < ql < "'" < qk (log k)
but p e r h a p s
~C
We
One
can
65
where
~C
depends on
C.
Perhaps
(I0) is a little too optimistic,
" m u s t " ( ? ) hold if k > exp(log k) I/2 Straus and I conjectured: primes.
T h e n for k > k 0
but (i0) certainly
w h i c h w o u l d easily i m p l y (9).
Let
Pl < PZ < " °"
there always is an
i < k0
be the s e q u e n c e of consecutive so that
2
(ii)
Pk < Pk+iPk-i
Selfridge with w h o m
"
w e discussed this p r o b l e m
strongly doubted that (ii)
is true, in fact he e x p r e s s e d the opposite conjecture. D e n o t e by
f(k) the n u m b e r
of changes of signs of the s e q u e n c e
Z Pk - Pk+iPk-i "
Perhaps
f(k) --o0 as
k
0 < i< k .
tends to infinity, this of course w o u l d be a v e r y considerable
strengthening of our conjecture with Straus.
I cannot even prove
Z A n old result of Tur~[n and m y s e l f states that Pk - Pk+iPk-i
li~n=sup f(k) = o0 .
has infinitely m a n y
changes of signs. Put
%
= Pk+l - Pk"
both have infinitely m a n y
T u r g n and I p r o v e d that dk+ 1 > d k
solutions.
We
and
dk+l < ~k
of course cannot prove that d k = d~+ 1
has infinitely m a n y
solutions.
W e further could not p r o v e that dk+ Z > dk+ 1 > d k
has infinitely m a n y
solutions.
It is particularly annoying that w e could not p r o v e
that there is n___oo k 0
(IZ)
so that for every
i > 0.
d! dko+i+l if i - O(rnod Z) and
Perhaps problems
dko+i < % 0 + i + i
w e overlooked a simple idea.
on consecutive p r i m e s :
if i ---l(mod 2)I '.
T u r i n has s o m e
Is it true that for every
d
very challenging and infinitely m a n y
n Pn ---Pn+l ( m ° d d)? Finally, in connection of our conjecture with Straus and Selfridge's doubts, the following question of Selfridge and m y s e l f m i g h t be of interest: be a sequence of positive density. l
ak+iak_ i ?
Let k
a I < a Z < ... and every
66
D o e s (13) hold if the density of a's is i? References I. Ruzsa,
O n a p r o b l e m of P. ErdBs,
Canad.
Math.
Bull. 15(1972), 309-310.
Ira. ErdSs and P. Tur{n, O n s o m e n e w questions on the distribution of p r i m e n u m b e r s , Bull. A m e r . Math. Soc. 59(1948), ZTI-Z78, see also P. Erd}Js, O n the difference of consecutive primes, ibid 885-889. P. ErdSs and A. R4nyi, S o m e S i m o n Stevin 27(1950), 115-126. P. ErdSs and K. Prachar, Univ. H a m b u r g 26(1962), 51-56.
p r o b l e m s and results on consecutive primes,
S~tze und P r o b l e m e
~ber
Pk/k,
Abh. Math.
Sea.
P. ErdSs, S o m e applications of graph theory to n u m b e r theory, Proc. second Chapel Hill conference on c o m b math. , North Carolina, Chapel Hill, N C 1970, 136-145. P. Erd$s, S o m e (1972), 91-95.
p r o b l e m s on consecutive p r i m e n u m b e r s ,
D. Hensley and Ian Richards, (1974), 375-391.
Primes
Mathenuatika 19
in intervals, Acta Arithmetica 25
67
9. Many
Some
extremal problems
extremal problems
explain w h a t I h a v e in m i n d < a
< n k(n)-
distinct.
in real a n d c o m p l e x n u m b e r s
on integers can be extended to real n u m b e r s .
consider the following p r o b l e m :
be a s e q u e n c e of integers.
Then
P r o b a b l y there is a
(Z)
c
maxk
n) 3/Z
Assume
that the products
a a. are all i j
< max
k (n) < w(n) + c I n 3 / /
og n) 3/Z
so that
(n)
=
w(n) +
but (Z) will not c o n c e r n us now. real n u m b e r s .
1 0
(1) is not yet solved for primitive sequences.
(I) one could characterize the s e q u e n c e s and a primitive s e q u e n c e
A s a first step to solve
n I < n Z < . ..
a I < . . . for w h i c h
for w h i c h there is
•
68
A ( Z nk) > E Z n k
for every
k = l,Z,...
T h e generalisations problems: every
Let
to real s e q u e n c e s
a I < a Z < ...
seem
be a s e q u e n c e
to lead to interesting diophantine
of real n u m b e r s
and assume
that for
i,j,k
(3)
Ika i - ajl >__ 1.
I cannot e v e n p r o v e that (3) implies
A(x) lira - -
A(x) = E l)
= 0,
x
a<x 1
O n e w o u l d guess that m o s t sequences
of the a s y m p t o t i c
properties w h i c h a r e valid for primitive
also hold if only (3) is a s s u m e d .
T h e only result is the following unpublished t h e o r e m that the
fact is not true for primitive s e q u e n c e s believe that m u c h assumed
of J. Haight.
a's are rationally independent a n d satisfy (3). T h e n of integers.
A(x)/--
/
of the difficulty will already be e n c o u n t e r e d
if the
lecture at Q u e e n s
College one m e m b e r
a's a r e
of the audience
S. Shapiro) a s k e d the following question w h i c h I h a d overlooked: be a s e q u e n c e
of real n u m b e r s .
Assume
J ]l ii i
Is it then
true
Let
(perhaps
1 < aI < ...
that
II aj lZl
j
for e v e r y pair of distinct choices of the finitely m a n y 13 .. J
x
In v i e w of Haight'sresult I
to be rational n u m b e r s . During my
and
Assume
--0. This in
non-negative
integers
~. i
that
(1)
~: 1 =A(x) 0,
there exist con-
such that
0
for all
O(z)~ = ~(z)$
is finite dimensional; (v)
stants
o(k)~
as a function of for
@
K = K • ~Kp-finite,
G/A spanned by the translates (iv)
y E GQ;
g e ~, a ~/A x
with
lal > A.
An automorphic
form
~
clal M
is called a cuspidal form if it also satisfies
83
the condition
f ~((~ ~)g)dx = 0 Q\~
for almost all
The space of cusp forms is denoted by
A0(~).
We also denote by
L2(G~Gt~,~)W/~ the Hilbert space of measurable functions
(i) (ii)
~(yg) = ~(g)
for all
p(z)~ = ~(z)~
g.
~
on
G/A such that
y E GQ
for all
z C Z/A
and
(iii)
f
l~(g)12dg < ~.
Z/AGQ\G/A
The subspace of by
L2(GQ\%, ~)
consisting of cuspidal functions is denoted
L20(GQ\G/A,~). It should be noticed that
subspace of
L2(GQ~G/A,~) consisting of
A0(~)
K-finite,~-finite functions
center of the universal enveloping algebra of unitary representation of
G/A in
GL2~)).
L2(GQ~%,~)
GL2~A)
if it occurs in some
Let
pC(g)
(~is
the
denote the
given by right translation.
An irreducible unitary representation of representation of
coincides with the dense
GL2(A) p~.
is called an automorphic
Recall that we have a decom-
position
pC = f~Sds@(~.~J ) J
into a continuous part and a discrete part.
An automorphic representation is
called cuspidal if it is equivalent to a discrete component of fact that any irreducible unitary representation of
GL2~A)
[ i0], p. 76); we write such a representation in the form
pC.
We use the
is factorizable (cf. ~ = ® ~ , where P P
runs over all primes including the infinite one, and for each
p,
~
P
p
denotes an
84
irreducible unitary representation
of the local group
Gp = GL2(Qp) o
The repre-
sentations which are of interest to us are those which have almost all their local components of class
1.3.
i.
Relation Between Modular Forms and Forms on Adele Groups.
When convenient we shall make use of the isomorphism between the complex modular variety
GL2(Q)\GL2(/A)/~XK
and
F\H
g = g0googf ÷
where and
K F
is an open compact subgroup of is the inverse image in
SL2(Q)
into
GL2(/Af) , /Af
SL2(Zg)
of
morphic forms on
K ~x
the ring of finite ideles
under the canonical injection of is identified with the subgroup
b2
The map that takes holomorphic
the subset
z = goo(i),
GL2(/Af) (cf. [ 3 ], p. iii); a2
given by
GL2(/A)
is given by
GL 2 OR) p ~ Kp
of
GL2~A )
cusp forms of f ÷ ~f,
Sk(N,~)
on
H
to autog = goo'gc in
where for an element
we put
~f(goog c) = (fl[goo]k)(i)gA(g c); here
gA
is the grossencharacter
following prescription:
~A = p ~ gp
canonical homomorphism from putting putting
(~ ~ ) ~
~p(a).
~Xp
to
The function
~f(%g) = ~f(g)
be an automorphic form on H,
of
for any GL2~A).
~x and
determined by Cp
(~ /N)X" ~f
~
according to the
is the pull back of gA
is extended to
is extended to all of
y C GL2(Q).
g
The new function
by the
~
Kp
GL2~A) ~f
by
turns out to
If we start with a real analytic form
then by the same prescription we get an automorphic form by letting
~f(g~gc ) = (f[[g~]o)(i)gA(gc).
by
f
on
85
We shall make full use of the one-to-one correspondence between the eigenfunctions of the Hecke operators on the space of holomorphic cusp forms or real analytic cusp forms which are new forms and automorphic representations (cf. [i0 ], p. 94, Theorem 5.19).
1.4.
Langlands' Euler Products.
We now review briefly Langlands' construction of Euler products from automorphic representations (cf.[~3 ], §2). The basic details for this construction can be found in Satake [ 32 ]. concern
G = GL2,
the following construction
particular we may take
For
p
elements and
K
Although most of the applications we make
G
to be a Chevalley group.
a finite prime let
Gp = G(Qp)
the maximal compact subgroup P
p = ~
we put
works for more general groups; in
G
= G~R),
be the group of G(~ ). P
the group of real points and
Qp-rational
For the infinite prime Koo the maximal compact
subgroup of G~, say corresponding to the involution associated to a Chevalley basis. As usual the adele group for all primes
p
G/A is the restricted direct product of the groups
with respect to the compact subgroups
subgroup of principal ideles in primes
p
is clearly a compact subgroup of
K = ~ K p P
GQ
is the discrete
taken over all
G/A. Let L2(GQ\~A)
be the space of
all square integrable functions on
GQ\~A
lations by elements of
Let ~
be the semisimple Lie algebra of
Let
be the Caftan subgroup of
K.
a Cartan subalgebra o f ~ . ~
G/A. The product
Kp.
.
Fix a Borel subgroup
consists of the
B
containing
~ • L2(G~G/A )
/
T
which are invariant under right trans-
T.
G
with
G
and + Lie algebra
The subspace of cusp forms L~(G~G/A)
with
~(ng)dn = 0
Gp
for all
g E G/A,
86
where
N
for all
is the unipotent radical of the parabolic P
except
G.
For a prime
subgroup
P
containing
p, which may be infinite,
H
B,
will denote P
the algebra of all compactly
supported regular Borel measures on
G
which are P
invariant under left and right translations
by elements
Kp;
(cf.
multiplication
is given by convolution
define the operator
%(~)
on
L~(GQ\G u /A)
%(~)~(g)
of the compact subgroup
[ 9 ], p. 278).
If
~ E Hp,
by
= f ~(gh)d~(h). G P
If
~
%(f) p
is the measure associated instead of
%(D)
all the measures
in
to a function
and consider H
f
f E LI(Gp)
as an element of
are absolutely
we sometimes write H . P
For a finite prime
continuous with respect to Haar measure.
P 2 L0(G~G/A)
The space is, for all ~i
p,
admits an orthonormal
an eigenfunction
generate an automorphic
%(~)
representation
We consider an element morphic representation
of
~
basis
~i,~ 2 .....
for all of
~ E Hp;
to it.
the translates
~ = ~
For a measure
~i of
i ~ .
/~GI^ which we denote by
of this basis and let
that corresponds
such that each
be the auto-
~ E H
we let P
~(~)~ = ~ ( ~ ) ~
and observe that the map
~ ~ Xp(D)
gives a homomorphism
of
Hp
into the complex numbers.
Let us now recall how all such homomorphisms into the complex numbers arise. Borel subgroup
B
Observe that, since T /T A K P P P
containing N \B PP
Let T ~d
N
be the unipotent
~
to
T , P
determines
and
any homomorphism a homomorphism
B
w:
% can> N \B --+ T /T n K w-w-+~. P P P P P P
P
of
Tp = T ( Q p ) . w
of B
into the P
complex numbers which we again denote by
H
radical of the fixed
N p = N(Qp) ' Bp = B(Qp)
put
is isomorphic
into the complex n u ~ e r s
of the Hecke algebra
87
If of
ad.b
b
belongs
to ~ ,
to
B
let
D(b)
the Lie algebra of
be the
N.
determinant
Since
G P
can be written as a product
bk
of an element
= B K , P P
b E B
of the restriction any element
g E G P
and an element P
k E K . P
Set
~w(g) = w(b)ID(b)[½.
The function
~
is well defined and any other function
w
(l.1)
on
G
satisfying
p
~(bgk) = w(b) iD(b)i½p(g)
for all
b,g
satisfying e H
~
and (I.i)
define
P
k
is a scalar multiple
are parametrized
~(~)~w
of
~w;
by elements
in fact all the functions
w ~ Hom(Tp/TpNKp,
~).
For
by
(%(~)Pw)(g)
= / ~w(g h)dD(h)" G P
The function scalar
%(~)~w
Xw(~).
satisfies
The map
and all homomorphisms
(i.i)
~ ÷ Xw(~)
of
H
and so
%(~)~w = Xw(~)~w"
then defines a homomorphism
which are continuous
for some
of
H
P
to
~,
in the weak topology are
P obtained in this way. a
u
The homomorphism
in the Weyl group so that
Suppose ;
p
is finite.
there is a homomorphism
Xw
equals
w(t) = w'(t q)
Let from
L
Xw,
for all
if and only if there is t E T . P
be the lattice generated by the roots of T /T A P P
K
or P
from
T
to
CL = Hom(L,2Z)
P
so that
] ~ ( t ) I = p%(t)(~)
if
~
is a root.
Here
~
is the character of
T
associated
to
~.
If
~
is
88
a root let
~
be the coroot attached to
~.
Let ~l,~2,...,~n be the simple roots
and
(Aij) = ~(~i,~i)j
be the Cartan matrix o f ~ .
The matrix
(~i,~.) (aij) = ( ~ )
is the transpose of The lattice
CL'
(Aij) and is the Cartan matrix of another Lie algebra
generated by the roots of a split Cartan subalgebra
can be identified with the lattice i n ~ in such a way that the roots of Also
CL = Hom(L,~)
generated by the eoroots
correspond to the elements
can be regarded as a lattice i n + ~ .
can in fact be regarded as the lattice of weights of Similarly,
~]R
may be identified with
the lattice of weights o ~ . algebra
c~
and let
an isomorphism
CT
o ÷ c
Let
CG
Hom(CLJR)
c~
of
el,~2,...,~n.
so ~
eL'
so ~ I R D L' D L,
be the Cartan subgroup corresponding to T
in
G
and
D CL D CL'. if
L'
be the simply connected group with
of the Weyl groups of
c~
~l,~2,...,~n
It contains
~
c~.
~.
is
Lie
There is
with that of
CT
in
CG
such that cu(%(t)) = %(ot),
If
w E Hom(Tp/TpAKp,
w(t) = ~%(g) CT
for all
associated to
%.
~), t.
t E T . P
then there is a unique point Here
% = %(t)
and
~%
g E CT~
is the rational character of
Thus associated to each homomorphism of
complex numbers is an orbit of the Weyl group in
so that
CT;
H
into the
P
or equivalently we may say
that to each such homomorphism there corresponds a semisimple conjugacy class in the complex group
CG~.
Let us now consider an automorphic representation
~ = ~
P
of
G/A in
89 2 \ L0(G Q G/A) which is unramified everywhere, i.e. each local representation a class one representation.
To an automorphic form
~
corresponds, for each prime
p,
of
finite let
{gp}
a homomorphism
be the conjugacy class in
Xp
CG~
in the space of Hp
into
~.
corresponding to
be a finite dimensional complex representation of
CG~
~p ~,
If
Xp.
is there
p
is
Let
r
and consider the Euler
product
~(s,~,r) = ~ det(l - p-Sr(gp))-l, P
the product being taken over all finite primes. that this product is absolutely convergent for shall see later on, in the particular case
Langlands Re(s)
has shown ([23 ], §3)
sufficiently large; as we
G = GL 2, Re(s) > 1
To the prime at infinity one also associates a
is enough.
F-factor.
Let
l
be the
homomorphism
T /T N K
which is such that in
~,
I~(t)I
= e l(t)(a)
every homomorphism of
Hom(L,]R)
+~IR =
if
T /T N K
~
is a root.
into
~
Since
L
is a lattice
is of the form
w(t) = e l(t)(X)
for some
X E ~.
Thus to every homomorphism of
an orbit of the Weyl group in
~
Hoo
into
~
there is associated
or a semisimple conjugacy class in
is the homomorphism associated to the automorphic form corresponding conjugacy class and let
dim. r det(l - r(X)T) = ~ (i - li(°°)T) i=l
9,
let
{X}
c~.
If
be the
90
be the characteristic product
~(s,Z,r)
polynomial of
r(X).
F-factor
that goes with the
is
F(s,~,r)
dim.r s-l.1 s-~. = ~ ~ 2 F(~). i=l
The Euler product associated the finite dimensional
to the automorphic
complex representation
L(s,~,r)
It is expected,
The
r
of
representation
CG E
and
is
= r(s,~,r)~(s,~,r).
and known in many cases, that
L(s,~,r)
satisfies
a functional
equation of the type
L(s,~,r)
where
E(~,r)
gredient of
is a complex number of absolute value r.
In some known instances,
automorphic
representation
exponential
factors that depend on
The delicate
~,
G = GL2,
the dual group
of the group
[ 2 ].
r
is the contra-
is allowed in the
may contain exponential
CG
CG~ = GL2(~).
is given in Langlands'
it suffices
factors
duals are the
In our particular
An excellent
[24 ],
to remind the reader that
their corresponding
A,C,B,D,E,F,G.
construction with many interesting variations Report
and
s.
A,B,C,D,E,F,G
types
1
ramification
g(~,r)
for our purposes
Chevalley groups of types
complex groups of respective
where
the number
construction
p. 25, in great generality; for
% = g(~,r)L(l-s,~,r),
introduction
case of
to Langland's
can be found in Borel's Bourbaki
gl
1.5.
The Functional Equation of Euler Products.
In the following we consider only automorphic representations of Let
~
be such a representation and for a finite p~ime
its local components.
The conductor
f(~ ) P
of
~
p,
let
~
P
GL2~A).
be one of
is defined by the following P
theorem of Casselman ([ 4 ], p. 302):
Theorem.
Let
~
be an irreducible admissible infinite dimensional P
representation of ideal
f(~p)
of
GL2(Q p) ~p
with central character
4.
such that the space of vectors
Then there is a largest v
with
~p((~ bd))V = ~(a)v
for all
(a c b) d C F0(f(~p) ) = {(a bd) E GL2(ZEp):
is not empty.
c - 0 mod f(~ p )} '
Furthermore, this space has dimension one.
We will say that a local representation The global conductor
f(~)
~
P
is ramified if
of an automorphic representation
f(~) = ~
~
f(~ ) # ~ • P P
is defined by
pordpf(~p),
P where the product runs over the ramified primes.
The construction of the Euler products associated to automorphic representations of
GL2~A)
can be done in various ways (cf. [ 2 ],[ i0 ]).
Here
we follow a combination of the method presented in Gelbart ([ i0 ], p. 113)
with
the method of Langlands
described in
§1.4.
First we consider the unramified
92
situation.
If
p
is a finite prime and the local representation
~
P
belongs to
the principal series then it is parametrized by two quasi-characters of x Qp: ~l(X) = Ixl sl, ~2(x) = IxlS2; if space ~(~I'~2) -
~0
is any
K -invariant function in the P of all locally constant functions @ on G such that P
q~((~l t2) g ) = ~l(tl)~2(t2) I
for all coset
]½#~(g)
is the characteristic function of the double
tl't2 @ QX and if T P P Kp(P 01)Kp' Kp = GL2(2Zp)
then the convolutions
@0*Tp(g) = / ~0(xy-I)Tp (y) dy G
P
= p½(pSl + pS2)@0(g).
To such a local representation we associate the conjugacy class
{gp}
in
GL2(~)
which contains the matrix
~p = (~sl ~s 2)
and to a finite dimensional complex representation
r
of
GL2(~)
we associate
the local factor
Lp(S,~p,r) = det(l - p-Sr(~p))-l.
To this local factor there corresponds a trivial root number the quasi-characters
~i
and
~2
g(~ ,r) = i. P
are both ramified then we put
Lp(S,~p,r) = 1
and the root number is taken to be, when
r = r2
the standard 2-dimensional
If
93 representation of
GL2(~) ,
g(~p,r 2) = W(~I)W(~2), where ~i"
W(~i)
is the root number of the local Tate zeta function associated to
If only one of the
~i' say D2'
is ramified, we take for local factor, when
r = r2, 1
Lp(S,~p,r 2)
l-~l(p)p -s and the root number is taken to be to the special representation and be
1
and
C(~p,r2) = W(~2). ~i
~(~p,r 2) = W(~I)W(~2).
is ramified then
Otherwise, if
=
Lp(S'~p'r2) and
g(~p,r 2) = W(~2).
representation
~
If
p
= ~(~i,~2)
L (s, ~ ,r 2) = where
%'i = -r.1 - m.l if
g(~ ,r 2) = i 2.
If
~
If
~i
~p = ~p(~l,~2) Lp(S,~p,r 2)
belongs
is taken to
is ramified, we put
1 l_Dl(p)p-S
is the infinite prime then for a principal series we put -½(S-%l) F S-%l) -½(s-X2) s-% (~ r(~J~)
~i(x) = Ixl ri sgn(x) mi.
= ~(~i,~2 )
For the root number we take
is a discrete series representation then
L (s,Z ,r2) = ~ - ½ ( S - % l ) r ( ~ l ) v - ½ ( s - % 2 ) r ( ~ - ~ )
where
%1 = -Sl
number we take
Let
and
%2 = -Sl - 1
if
~i(x) = Ixl si sgn(x) ni.
For the root
~(~ ,r2) = iSl-S2 +I.
S
representation are unramified.
be the special set of finite primes ~
p
for which the local
= p(~l,~2) is a special series representation and DI,~2 P We define the special conductor of ~ and the special root
number, respectively, by
f0(~) = ~ p , p~S
g0(~,r2 ) = (-1) ISl]-~l(p) , pES
94
where the second product is taken over all the quasi-characters in the special representations
~
= p(~l,~2)
for
~i
that appear
p E S.
P
The global root number associated to an automorphic representation of
GL2~A)
and the standard
2-dimensional representation
r2
of
GL2(~) is
given by
g(]~,r2) = ]-~ g(~p,r2). P
The Euler product associated to
~
and
r2
is
L(s,~,r2) = -~- Lp(S,~p,r2). P By Jacquet-Langlands
([ 16 ], p. 350, Theorem ii.i) we know that if
cuspidal automorphic representation of the Euler product
L(s,~,r2)
GL2~A)
~
is a
with central character
4,
then
represents an entire function, is bounded on vertical
strips of finite width and satisfies the functional equation
L(s,~,r2) = g0(~,r2)f0(~)l-sg(~,r2)f(~)½-SL(s,~,~2 ),
where
1.6.
r2
is the twisted contragredient representation
~-ir.
Some Examples.
The Euler Products of Hecke. a Dirichlet character of
(~/N) x
a holomorphic cusp form of weight
k
Let
k
and
and assume on the group
N
be positive integers and
(-i) k = 4(-1). F0(N).
Suppose
eigenfunction for the Hecke operator
fiTp =
oo co ~ a qn + ~(p)pk-i ~ anqpn ' n=l ~ n=l
Let
p ~ N
f(z) f
is an
be
95
and of the operator
U
P oo =
fIUp
~ a
qn
n= 1 pn
with the corresponding eigenvalues being
,PIN,
a . Define the zeta function of P
f
by
co [ a n -s n
¢(s,f) =
n=l
= ~
To the cusp form f(~) = N
f
(I-app-S)-I p~N (l-ap p-s+~(p)pk-I-2s)-I"
corresponds an automorphic representation
~f
of conductor
whose Euler product is none other than
S-%l ~s-X2 s-%2 k-l, s-X~ _ ~i_Xi~ L(s,~f,r2) = ~ - ( ~ ) F ( - - 2 ~)~ ~F(~)~(s+-f),
where
k-I
XI
2 '
k+l
%2 = -
2
i
The functional equation is
L(s,~f,r2) = e(~f,r2)N½-SL(s,~f,r2) ,
where ~A
Ig(~f,r2) i = I.
Incidentally, when
A
is the Ramanujan modular form and
is its associated automorphic representation then
g(~A,r2) = i,
f(~A ) = i,
and
L(S,~A,r2) = 2 ( 2 ~ ) - ( s + ~ ) F ( s + ~ )
~ T ( n ) n - S - ~ -. n=l
This is an example of an Euler product associated to an automorphic representation
96
~A =
®P7 P
component
which is unramified everywhere. ~
Also in this example the infinity
is a member of the holomorphic discrete series.
The Euler Products of Maass. field of discriminant
d.
Let
CK
the two element Galois group of
Let
K = Q(~)
be a real quadratic number
be theidele class group of
K/Q.
Let
E
K
X((~)) =
To each rational prime
p
and
G = {I,T}
be a fundamental unit of
the real Dirichlet character associated with the extension an unramified grossencharacter of
K
K/Q.
Let
K
and X
be
whose value at a principal ideal is
4
~ik/l°g
we attach a conjugacy class
{gp}
in
GL2(~)
with
det(l 2 - Tr2(gp)) = 1 - ap T + @(p)T 2
where the coefficients
a
are defined by P
ap = k(~) + X ( ~ )
if
(p)=~.~T
ap = 0
if
(p) = ~ .
To the infinite prime we associate the eonjugacy class
X
~ GL2(~)
whose char-
acteristic polynomial is
. k~
.2
2
det(l 2 - Tr2(X ) ) = i + (l--~--~g g) T .
The resulting Euler product
s-~ l s-~ z L-S'~K'r2-() = ~- - - 2 - - r ( ~ ) ~ - ~ F ( s - a 2 ) 2
where
~ det(l-p-Sr2(gP))-l' P
97
~ik log E
%1
and
%2
~ik log
satisfies the functional equation
L(s,~K,r2)
= E(~K,r2)d
½-s
L(l-S,~K,r2).
This is the Artin-Hecke L-function associated with the 2-dimensional of the Weil group
WK/Q
obtained by inducing the character
Recall that the Weil group
WK/Q
of the pair
K, Q
X
from
representation CK
to
WK/Q.
is the group extension
i -+ C K ÷ WK/Q ÷ G ÷ i
obtained from the distinguished L(S,~K,r 2) o K = @pOp
H2(G,CK ) .
is in fact the Euler product of an automorphic representation whose infinity component is a principal series representation.
automorphic form associated to Tne map
generator of the cyclic group
~K ÷ OK
§2
[ 26 ].
in number theory and has been analyzed
we will consider other examples of
of
Maass
([ 16 ]).
with automorphic representations representations
is one of those considered by
is of great significance
in depth by Langlands
In
oK
The
GL2(~)
of
GL2~A)
other than
r 2.
Euler products associated
and finite dimensional complex
g8
§2.
2.1.
Rankin's Convolution Method.
The Ingredients.
Let
N
be a positive integer and let
gruence subgroup. with
s
Fix an eigenvalue
pure imaginary or purely real between
character of of functions
~/QX
of conductor dividing
~
2 Lo(GQ/G/A,g) ,
in
Casimir operator
A
with
K 0 = p]~< K P g
P
-i
N.
G = GL2,
for all
and
the natural
and
g
W (N,%)
we have
K P = {I~ bd) E GL2(~p): g
Let
be a grossenthe subspace
such that under the action of the
g E GL2~A), r(@) E K
restriction of
i.
Denote by
at the 'infinity' component
~(gr(@)k 0) = C(ko)~(g) where
be the usual Hecke conl-s 2 of the Casimir operator; assume % = 4
%
Fo(N)
to
and also
= S02(I~) and
c ~ 0
K . P
A~ = %~
mod. N}
W (N,%)
k
and
E K0, g = ~
Cp,
has the structure of
a finite dimensional Hilbert space with the inner product
(~i,~2) =
/
~i'~2
dg.
Z/AGQ\ G/A
The natural isomorphism
Z/AGQ\ ~A/K K0 $ F 0 (N)\ SL 2 (~)/S02 OR)
gives a correspondence between functions on the group and functions on the upper half plane:
~(g) ÷ f(z)
adele group element
g
with
z = g~(i),
where
at the infinite prime.
g~
is the component of the
Under this correspondence the
above inner product is the relative Petersson inner product
(f(z),g(z)) =
/
f(z)g(z)d~,
D0(N)
is the
SL2-invariant measure on the upper half plane and
where
d~ = y-2dxdy
D O (N)
is a fundamental domain for
F0(N).
Hecke operators
T
P
and
T
P
acting
99
on the space
W (N,~)
are defined as usual ([ i0 ], p. 88 for the adele setting
and [26 ], §4. for the classical case). W (N,%) and
generated by functions
dINN~; let
W+
Let
W- (N,~) s
g(dz), where
g(z)
be the subspace of
is an element of
be the orthogonal complement of
Ws(N,%)
in
We(N',%)
WE(N,%).
In the following the elements of Wc(N,%) will be viewed as functions on a b the upper half plane. Let A ( v d~ ' 1 < v < D0(N) = N p ~ N (i +--i) run over v Cv v P a representative system of elements in F (i) which correspond to a complete set o a of inequivalent rational cusps let
be t h e s m a l l e s t
KV
Fo(N).
v = O v
rational
for the group
v
number
K
A simple calculation shows that
1-r 2 = 4 ,
cusp
c
p
For each such
Kv(C~,N) = N.
Now if g
in
o
v
1 K)A-I E Av(0 1 v
for which the matrix
then the Fourier expansion of a function
z = x + iy
W (N,~)
and
about the
has the form
gl[p]o(Z) = ~
½
ap(n)y Kr(
K
is the modified Bessel function.
r
expansion of a function cients
a(n)
g(z)
) exp .--~---~, P
If we want to consider the Fourier
only about the cusp at infinity, there the coeffi-
will be written without any subscript except possibly to denote
their dependence on the function definition:
~2~inx~
2 ~K P
n#0 where
Fo(N).
a new form
g(z).
f @ Ws(N,%)
For convenience we introduce the following
is a non-zero element in
a common eigenfunction of all the Hecke operators
T
with
W~(N,%)
which is
(p,N) = I;
the
P function
f(z) E Ws(N,% )
the cusp at infinity has
Remark 2.1.1.
is said to be normalized if its Fourier expansion about a(I) = i.
As was already pointed out in
~i,
a new form
in the above sense corresponds to an automorphic representation GL2~A Q)
whose local component
~
~ = ~
fEWs(N,% ) P
of
at the infinite prime belongs to the principal
series and whose restriction to the maximal compact subgroup
02~R)
is trivial.
Rankin's convolution method, which we explain below, can be applied also to automorphic representations where the restriction of
~
to the maximal compact
100
subgroup is not trivial; we will not consider here this case in order to avoid complications of notation that result from having to introduce a Bessel function whose structure is more complex than that of the modified
Remark 2.1.2.
If
f E Ws(N,% )
is a normalized new form whose Fourier
expansion about the cusp at infinity has coefficients Dirichlet series
~(s,f) =
~ a(n)n -s n=l
Ks(Z).
{a(n): n C ~ },
then the
has the Euler product expansion
~(s,f) = q ~ N (i- a(q)q-S) -I p ~ N (l-a(p)p -s+£(p)p-2s)-l.
As in
§1.4,
if we put
L(s,f) = ~-½(S-%l)F(S-%l)~-½(s-%2)F(S;~2)~(s,f), 2
with
l-s %1 = - r +
and let then
~
(-i__~) 2
%2 = r +
'
be the automorphic representation of
L(s,f) = L(s,~)
is the Euler product
2-dimensional representation of
GL2(~) ,
I-E (-i) 2
GL2~A)
L(s,~,r2),
associated with with
r2
f,
the standard
and it satisfies the functional equation
L(s,Z) = £(z)N½-SL(I-s,~),
where
E(~)
is a constant of absolute value
representation
If
k
1
and
~
is the contragredient
~(g) = ~(g)-I (g) ([i0 ], p. 116).
is a positive integer and
%
k(k-l) 2 ,
denote the space of holomorphic cusp forms of type properties of new forms in
HE(N,%)
we also let
{Fo(N),k,E}.
H (N,%)
The concept and
which we shall use in the following are
developed at great length in Winnie Li's article [25 ].
Here we recall the well
101 known fact ([ i0 ], p. 91) morphic representation
that a new form in
~ = ~p
of
H (N,k)
GL2~AQ)
corresponds
to an auto-
whose component at infinity belongs
to the holomorphic discrete series.
Another important ingredient that is used in Rankin's convolution method is the theory of Eisenstein series for Kubota's book [ 17 ]
Let
rp
be one of the
SL2(~)
stabilizer in
Yo(N)
o = (ac b)d
j(o,z) = cz + d. Im(z) = y.
Let
If k
go(N)
rational cusps of
which caries the cusp of the cusp
in
SL2~R )
rp, i.e.
and
z = x + iy
z
i~
into
Fo(N) rp.
and let
Let
Fp
p
be
denote the
Fp = {4 E Fo(N): o(rp) = rp}.
For an
a complex number, we write as usual
is a point in the upper half plane we put
be a positive integer and
We extend
X
Po(N);
denotes a complex variable.
s
The basic reference here is
from which we borrow freely the following results.
an element of
element
~ (N). o
to a character of
Fo(N)
X
a character defined modulo
by putting
X(o) = X(d)
To the data
{s,N,x,k}
for
N.
o = (ac bd) E
we associate the
Eisenstein series ([17 ], p. 63)
Ep(z,s,x,k ) =
~
X(O)~(p-lO'z)
OEF hE P
i)k(imp-lo(z)) s,
"lJ(p-lo'z)
where the sum runs over a complete set of coset representatives modulo
r . P
r%
has the form
~p,XyS 6p,%
r = r (N) o
We recall that the constant term in the Fourier expansion of
about the cusp
where
of
+
~p, ~(s,X)kyl-S
is the Kronecker delta function,
kl
(-i) 2Z2F (s) F (s-1) ~p, %(s'X)k = k k
r (s+~) r (s-y)
~p, %(s,X),
E
P
102
and
~p,Tt( s, X) =
X(Pd%-l) le1-2s,
~ (* * O= c d )
where
• d * ) (c
runs over a complete
which are inequivalent
plete set of inequivalent
If
cusps for
¢(s,x)
that this square matrix,
the analytic
in Foo = { (0 1 ): n 6 2g }
modulo the group
by right and left multiplication.
Observe
set of coset representatives
continuation
p
and
F (N) o
under its action
run independently
then the constant
over a com-
term m a t r i x is
= (¢p,X(s,X)k).
w h i c h plays an important
of the Eisenstein
Here w e shall use the following
~
p-lro(N)k
of
series
role in the theory of
Ep(Z,S,X,k) ,
theorem whose proof is identical
has
~o(N)
to that in
rows. §6.2
in Kubota's book:
T h e o r e m 2.1.
If the rows in the column vector of Eisenstein
series
~(z,s,x,k ) = t(El(Z,S,X,k ) ..... E~o(Z,S,X,k)
have the same order as the rows in the constant functional
term matrix
~(s,x) ,
then the
equation
~(z,s,x,k ) = ~(s,X)~(z,l-s,x,k)
holds.
Remark 2.1.3. argument
similar
the constant
If
X
is the principal
to that given by Kubota
character and
([17 ], p. 45)
term m a t r i x has a simple pole at
s = i.
k = O,
then an
shows that each term in In all other situations
103
Ep(Z,s,x,k)
s = i.
is regular at
Ep(Z,S,X,k)
not so then the residue of independent of
z
To see this we simply observe that if it were
and also
at
s = 1
would be at the same time
X-automorphic and this is impossible.
After these preparations we are now ready to look at a typical example of Rankin's convolution method.
Theorem 2.2.
Let
~
and
~'
be automorphic representations of
associated respectively to a holomorphic new form 11
k(k-l)2'
=
Wg2(N2,12) modulo
k
i- r 2 2
12
induced by
about each cusp
in
HgI(NI,I I)
an integer, and to a real analytic new form
with
N
f(z)
r
P
gl~2 . of
Let
N = ~.c.m. (NI,N 2)
and let
Suppose the Fourier expansions of
F (N) o
g(z) X
GL2GA Q) with
in be a character
f(z)
and
g(z)
are given by
2~inz fl[p]k(Z ) = n~lap(n) e
0},
th en
Sp E
[
P-IODo (N)
o E ro(N)/r p
up to a set of
d~-measure zero, where the sum runs over a complete set of coset
representatives
of
Y (N) o
Remark 2.1.6. be applied to
modulo
The above congruence identity between regions, which will
F -automorphic P
functions,
as an excercise for the reader.
Lemma 2.4. have for
Re(s)
F . p
is relatively easy to prove and is left
(See [ 30 ], p. 367).
With the same assumptions and notation as in Theorem 2.2. we
sufficiently
large k
/ y2fl [p]k(z)g---r[P]o(Z)ySd~ S
K = K 7TI(TZ~ ~] p 4#
where
k-i 2
P
S+--
£(s+½(k-l)+r)F(s+½(k-D+r) £ (s +k)
co ~ ap(n)~ p (n)n_S_½(k_l) n= 1
d~ = y-2dxdy.
Proof.
We multiply the Fourier expansion of
conjugate Fourier expansion of
gl[p]o(Z)
fl[p]k(Z)
by the complex
and integrate the product with respect
106 to
K K in the interval " ~[--~'~]' zz
x
where
K = Kp,
to obtain
K
S
fJ[P]k(z)g~[P]o(Z)dx
K 2 2~ny
oo
= K ~ a (n)bp(n)y½Kr(~)e n= 1 O
where y~+
z = x + iy, 2
S
y > 0.
K
We now multiply both sides of the above equality by
and integrate the resulting expression over the interval
respect to
y.
We evaluate explicitly
[0, ~]
with
the Bessel integral by using the well
known identity ([ 27 ], p. 92)
S0e-aXx~-iK valid for
Re(~+~) > 0
(ax) dx = ~½(2a) -~ F(D+~)F(~-~) r(~+½)
and a real; thus we get o~k+s_2
K
SY
S~-
fJ [P]k(z)g~[P]o(Z)dxdy
0 2
n=l~ap(n)bp(n)n
F(s+k)
"47
= Lp(S,~ x ~').
The interchange in the order of summation and integration is justified for
Re(s) > Oo
by the fact that
gl[p]o(Z)
and
fJ[p]k(Z)
are
0(y c) (resp.
--C v
0(y
))
constants
uniformly in c
and
c'.
x
as
y ÷ ~
(resp. as
y ÷ 0)
with suitable positive
107
Proof of Theorem 2.2.
By lemmas (2.3)
and
(2.4)
we have
k Lp(S,~×~') = f y2fl[P]k(z)g-TT~o(z)ySd~ S
P k
=
~ OEFo(N)/~
~
=
f i Y2fI[P]k(z)g-TT-PTo(z)ySH~ p- ~Do(N )
/
k y2fl[P]k(z)g-TT~o(z)ySd~o(p-lo).
a~ro(N)l ~ Do(N) We now use the transformation formulas
f ~-~-$-~J raz + b~ = ~l(d)(cz + d)kf(z)
and
raz + b~ g~c-~-~-~j = E2(d)g(z),
which hold for any
~ E F (N),
to obtain
o
k (Imz)2f][p]k(z)g~[p]o(Imz)Sd~o(p-lo)
= (Imz)~f(z)g(z)gl(~)~2(o)l j(p-lo'z) l)k(imp-lo(z))Sd~,
~lj(p-lo,z)
where
j(T,z) = cz + d
if
T = (c d)-
This change of variable applied to the
last integral gives k Lo(s,~X~') = /
y2f(z)g(z) Ep(z,s,x,k)d~ ,
D
(N) o
where we have put
108
Ep(Z, s,x,k )
X(o) lj (p-lo,z)] k (Imp -I o(z) )s.
~
=
oE £o(N)/rp
Now the functional
(s,x)]L(1-s,~TX~'),
which is what we wanted to prove.
2.2.
The Constant Term Matrix for the Eisenstein
It is possible to go further than Theorem about the structure of the constant
Series of the Group
Fo(N).
(2.2) by using information
term matrix for the Hecke groups
£o(N).
The
results that we need are already available in the literature and are due to Orihara
[ 28 ], §3.
is more convenient
Let
N
We now proceed to describe these results using a notation that for our purposes.
be a positive integer and let
FN
denote
the principal
congruence
109
nl... subgroup
of level
N.
Let
N = p]
n1 p]
be the faetorization
of
N
into
n.
distinct
primes.
We put
N = NiP i i,
1 < i
"
L(s,~X~')
where
(f,g>
_
+ ( f'g)~ +
I c(n)(s-l)n, n=l
is the Petersson inner product of f and g
and
~
is a modified
inner product defined by
~ = 6(c -~io$ 2) < f,g) _ 2 ~ / f(z)g(z) log (y61A(z)l)d~, D(F)
where
A(z)
is the Ramanujan modular form.
Proof.
First observe that because of the parity assumption
(f(z) K)(g(z) K) = f(z)g(z)
we have that
SF = { z = x + i y :
Let
F
= {(
n
i): n C 2Z }
F = SL2(~) /{+12} ;
and let
D(F)
a(-n)b(-n) = a(n)b(n).
Ixl j ½, y ~ 0).
be the fundamental domain for
then we have as in Lemma (2.3)
sr s
On the other hand we have for
Re(s)
Let
~ a e r/r
the formal congruence identity
aD(r).
sufficiently large
/ f (z)g (z)Y sd~ SF
½ = / (/ f(z)g(z)dx)yS-mdy 0 -½ .t 2~i(m-n)xl ~ s-2_ = / ( ~ ~ a(m)b(n)YKr(2~[mly)Kr,(2~In[y)j e axjy ay 0 m n -½
oo
= m~0 a(m)b(m) 0f
Kr(2~]m[y)Kr'(2~Imly)yS-ldy"
The i n t e r c h a n g e of t h e o r d e r of s u m m a t i o n and i n t e g r a t i o n
is justified
by t h e f a c t
116
that uniformly y ~ ~
in
x
both
(resp. y ÷ 0)
f(z)
and
g(z)
0(y c) (resp. 0(y
are
for some positive constants
We now use the well known identity
c
and
-e'
))
when
e'.
([ 27 ], p. 102)
oo
/ K (C~t)K (~t)t-Pdt 0 P e~P-I 2P+2F (l-P)
which is valid for = 2~ny
and
Re(s) > 0
p = l-s
and
Re(I-o±~±V)
> 0.
We apply this identity with
to obtain
ff(z)g(z)ySd~ SF -s
s+r+r'
~.s+r-r'
s-r+r'
.s-r-r'
r(---f~)~t~)r(T)r~
~)
a(n)b(n)n -s
4F(s)
n=l
= L(s,~ ×~').
On the other hand using the congruence
L(s,~X~')
=
identity for the region
~
/ f(z)g(z)ySd~ D(F)
act/to / ac F/F
From the automorphy property of
f(z)
f(z)g(z)ySd~oo
and therefore,
and
f (z)g(z)Y sd~°(7-
D(F)
g(z)
we get
= f(z)g(z)(Im~(z))Sd~
making the change of variable
s+l s ÷-2'
SF
we have
117 L.l+s t-~,~x~')
= /
f(z)g(z)E(z,s)da,
D(F) where
E(z,s) =
~
(ImO(z)) ½(l+s)
O e F/F
is the well known Eisenstein series associated to tional equation for
E(z,s)
F.
We recall that the func-
is
A(s) E(z,-s) E(z,s) = A(s+I)
where
A(s) = ~
-½S
representation for
S
F(~)~(s).
This functional equation applied to the integral
L(s,~ x ~')
gives
l+s _
L(--Z-,nx~')
A(s)
= A(s+l) L(
7~
,~x~').
We now recall that the first Kronecker limit formula states ([20 ], p. 273)
E(z,ms-I)
3 ~
i +6 s-~ ~(c-log
2) - 1
log (y61A(z) i) +
7 e(n)(s-l)n, n=l
where
oo
A(z) = q-~-(l-qn) 24,
q = exp 2~iz,
n=l
is Ramanujan's modular form. is regular for
Re(s) > ½
From this expansion and the fact that
except for the pole at
s = 1
E(z,2s-l)
([17 ], p. 44)
the
claim in Theorem (2.7) follows easily.
Remark 2.2.3.
Various other possibilities
convolution method suggest themselves;
for developing Rankin's
among these three are noteworthy of mention.
First the result of Ogg-Winnie Li ([25 ], p. 313)
gives an exact functional
118
equation for an Euler product related in a simple way to ~'
correspond to holomorphic cusp forms of levels
condition that if a prime in §2.2 below). is proved for weight
k
then
L(s,~ × ~') 1
when and
~
~'
when
~
and
N1
and
N2
qllN1
and
qiiN2 (see example 3
Secondly in Doi and Naganuma [ 8 ]
and level
§2.3 below).
qig.c.d (NI,N2)
L(s,~X~')
satisfying the
an exact functional equation
is associated to a holomorphic cusp form of is a real analytic cusp form (see example 5 in
The last, and perhaps most attractive of all, is the result of
Jacquet [ 14 ]
where it is shown that
L(s,~ × ~')
or rather a simple multiple of
it, for arbitrary automorphic representations n and ~' field, satisfies a functional equation.
of
GL2~),
k
a global
It appears that for applications to
arithmetic questions, the result of Jacquet promises to be of much significance, even in the case of
GL2~AQ).
In a future publication we will persue the problem
of making explicit in the case of
GL2~AQ) , Jacquet's form of Rankin's convolution
method.
2.3.
Some Euler Products and their Functional Equations.
In this section we will give several examples of Euler products that satisfy relatively simple functional equations.
Example i.
We begin by observing that under the assumptions made in
Theorem (2.7) together with the restrictive condition that g(z) iK = g(z)
we have, with
% = r
and
f(z) IK = f(z)
%' = r',
L(s,~) = ~ - ½ ( s - % ) F ( ~ ) F - ½ ( s + X ) F ( 2--)s+lii~(l-~pp-S'-l" I ) ( _~pp---s,-l) P
= L(I-s,n)
and
and
119
L(s,~') = ~-½(s-%')r(s2%----")~-½(s+%')F(S2%') ~ (l-~'p-S)-l(l-~p-S) -I P P P =
L(l-s,~').
The multiplicativity of the coefficients
a(n)
and
b(n)
give after an easy
calculation
oo 1 --a(n)b (n)n-s = ~(2s) -~-det(14-P Sr(gp)®r(gp )) i n=l P
1 , -s -i -- , -s -I ~-i -s -i - , -s -i - ~(2s) p~(l-~p~pP ) (l-~p pp ) (l-~p~pp ) (l-~p~pp ) •
Here
r
is the
2-dimensional representation of
is the conjugacy class in If we now define a ~'
GL2(~)
GL2(~)
containing the matrix
and
{gp} (reap. {g~})_
(0~P0_)~p (reap. (0~
F-factor associated to the automorphic representations
and the complex analytic representation
r@r
~p))0. ~
and
by
F(s,~,~';r@r) =
= ~-½(s-k-l') F (.s-l-k' T) -~(s-k+k').~s-k+k'.i ½
with the possible exception of a simple pole at
which occurs precisely when
say when
~
is equivalent to
Example
3.
L(s,~,~';r@r)
~
is the contragredient of
7',
that is to
~'.
This example, due to
Ogg and Winnie Li ([ 25 ], P. 313),
deals with two automorphic representations which may be ramified but whose conductors satisfy certain arithmetical properties. representation of conductor
N1
GL2~A~)
and
trivial central character; let GL2~A Q)
k,
and trivial character.
N2
product of all primes that for every prime
q q
~
be an automorphic
associated to a holomorphic cusp form of weight
representation of conductor
Let
7'
be another automorphic
associated to a holomorphic cusp form
that divide which divides
M
M
Let
g(z)
M = ~.c.m. (NI,N 2)
and
and for which ordqN 1 = ordqN 2. N,
f[[Vq ]k = ~qf'
ordqN 1 = ordqN 2 = 1
gl[v_M]kq = nqg
with
~2 = 2 q Nq = I,
k,
and
of weight N
is the Suppose
122
where
Vq
and
x~y,z
M
q x yq ) ' ~ = ordq M = (Mz
q
are integers satisfying
2~
x - yMz = q .
If we define a
P-factor
by
P(s,~,~';r®r)
4 = ~-2(s-%i)F(~), i=l
where
%1 = l-k, %2 = -k, %3 = 0, X4 = -1
and if we put
L(s,~,~';r®r)
= F(s,~,~';r®r) (1-~qnqq-S)-l~det(l.-p-Sr(g~ql II'N ) 8r(g')) -Ip P~M 4 p
v -i ~"(det(12-q-Sr(gp))-l~"'det(12-q-Sr(gq')) • ~ ' ( l - q -s ~q~q)
-I,
where
L(S,~) = P(s,~) ~ (l-Eqq-S)-i V -s -i -- -s -i qlNl P~Nl(l-Epp ) (l-Epp )
and
L(s ) =
(lqqS>1 V (l ppS) l(l%pS) -1 q IN2
are the Euler products of the conjugacy classes in
~
and
GL2(~)
P~N 2 ~'
respectively and
{gp}, { gp'} ,{ gp'} ,{gp,,}
that contain respectively the matrices
are
123
0 ~' (~p ~p) , (0p _~,) , [~q~ 0q~q) ' O,
Nlc
and
then
ImO(z) = icz+dl2, Y z = x+iy.
Thus
FN(Z,S ) = yS +
ySlmNz+nl-2s"
~ m>0 (mN,n)=l
Let
-I ~N(S) =
n>0~ n-S = p~N (l-p-s) (n,N)=l
Then we have
2~N(S) FN(Z, s) =
~ ImNz+n 1-2s m,n (n,N)=l
)~'t,~=+~l -ms Z ~(d) m,n
dln,N d>0
in
as
represent-
(c,d) = i.
that if
O(z) = az+b cz+d
F
F (N) are o
Recall
135
=
[ B(d>
diN
[
]mNz+n1-2s
m,n
= d~N~(d) d-2SG(~, s) ,
where (m,n)
~
is the Mobius function, the sum E' m,n different from (0,0) and
G(z,s)
is an Epstein zeta function.
=
runs over all pairs of integers
~' Imz+n1-2s m,n
We use the fact
E(z,s) = ySG(z's) 2~(2s)
=
Z
(Im°(z)) s,
o 6 F/F where series.
F
is the unimodular group
SL2(~ ) /{±12 }
and
E(z,s)
Thus we get
YS p ~N (1-p-2S)FN (z,s) = N-Sd~ND(d)d-SE(~,s).
Let
E(z,S,Fo(N)) =
[ (ImO(z)) ½(s+l) e F (N)Ir o
and
E(z,s,F) =
[ o6F
(ImO(z)) ½(s+l) Foo
is its Eisenstein
136
denote respectively r (N) o
and
the Eisenstein series for the cusp at infinity for the groups
F-(observe the change of variable
s ÷ ½(s+l)!).
We then have the
identity
E(z's'F°(N))
= pIN~(l-p-l-s)-i d~N p(d)(Nd)-h2(I+s)E(~'s;F)"
The well known Fourier expansion of easy change of variable
E(z,s,F) = y½(l+s)+
where
Os(n)
A(s) = ~
-½s
([17 ], p. 46)
in the simple
can be put, after an
form
l~1½s .y½K~ =s (2~Imly)exp(2~imx)'
is the sum of the
and
E(z,s,F)
A(s) ½(l-s), ~ 2 Os (Iml) A(~ y tm$ 0 ^(s+l)
s F(~)~(s)
E(z,S,Fo(N))
s+l s +--~--,
s-th
powers of the positive divisors of
is Riemann's Euler product. E(z,s,F)
we substitute
n
and
In the identity relating
this last Fourier series to obtain,
after rearranging the terms involving the Mobius function
p(d),
E(z,s,F o(N)) = y½(l+S)+c(s)y2(l-S)+m~0Cm(S)y2K½s(27[m[y)exp(2~imx),
where
A(s) ~I(N) c(s) = A(s+l) }s+l(N)'
~a(N) = N a T ( 1 - p PIN
and the coefficients
c (s) m
are all holomorphic
To obtain the residue of several ways.
-a )
E(z,S,Fo(N))
in the region
at
s=l
The easiest is to evaluate the residue of
interesting way is to appeal to Kronecker's
Re(s) > 0o
we can proceed in c(s).
A somewhat more
limit formula ([20 ],p° 273)
for the
137 Eisenstein series
E(z,s)
which in a neighborhood of
s = 1
can be written as
1 12 k 2) 6 E(z,s,F) . . . . .s-i + - 7 (Y - log 2 - log y2]q(z)] + %T
and then substitute into the identity which relates
E(z,s,F)
0(Is-il) to
E(z,S,Fo(N)).
We then obtain
E(z,s,Fo(N)) = U (l-p-l-s)-i ~ p(d) (Nd) -½(s+l) pIN diN 6 x (~- s_--ll+ (y - log 2 - log (N--Xd)½[~(~)I2) + 0(Is-ll)),
and this is
= 6.
1
.i
~T N g ( l + p I-) s-i
We can even compute the constant
A(z)
+ A(z) + 0(Is-ll)
in the Laurent expansion by using the
appropriate terms in Kronecker's limit formula. to obtain, again after a change of variable
We put together the above results
s+l ~ + s,
oo
L(s,~X~)
3 < f,f> = ~'Ng(l~)
A comparison of the poles of
.i + !0an.(S_l)n" s-i n
E(z,s,Fo(N))
and using the identity
(4~) -Sr (s~--r)r (s~--r)F (2) 2 la(n) 12n -s = L(s,~X~) 4F(s)
shows that
r
n= 1
cannot be real, and in particular
the F-factors one obtains
r # ½.
Therefore dividing by
138
n=l
la(n) 12n-S = 3__ • ( f,f ) 73 F(½+r)F (½-r)
1 Np~N(l+p-l)
_i_l + [ bn(S_l)n" s-i n=0
A standard application of the Wiener-Ikehara Theorem to the above Dirichlet series gives
la(n) l2 = ~ . n<x
( f,f > F (½+r) F (½-r)
• x
+
o (x).
NyN(I+p-I)
This completes the proof of Theorem 2.8.
The Cauchy-Schwarz inequality and the above asymptotic estimate give the following corollary.
Corollary 2.9.
With the notation and assumptions as in Theroem 2.8.
have
la(n) l ½;
c(n) > 0
and
and
First we recall Landau's Theorem [ 18 ]:
are real numbers satisfying the inequality
(II)
L(s,~ x ~),
we
139 co
Z(s) =
[ c(n)n -s n=l
is absolutely (III)
convergent
The function
for
Z(s)
Re(s) > B
and represents
has a meromorphic
plane and in each fixed strip
o I < o < 02
there a regular function;
continuation
to the whole complex
it has at most a finite number of
poles; (IV)
for some
A > 0
co
F(~I+BIS)...F(~4+B4s)Z(s)
= F(yi-61s)...F(y4-64s ) [ e(n)(An) s, n=l
the last sum being absolutely convergent (V)
Re(s) < 0;
Z(s) = 0(e Yltl)
for large (VI)
for
Itl
and some constant
for some constant
y = Y ( O l , o 2)
in any strip
o I < O < 02;
B > 0
le(n) InB = 0(xB(log x)B). n<x
Then we have
c(x) =
~ c(n) = a(x) + 0(xK(log x)g), n<x
where
g = max (B,m(B)-I),
m(B)
is the multiplicity
of the pole of
Z(s)
at s = B,
2D-I K = B" 2N+I'
and
R(x),
(x > 0),
in the strip
is the sum of the residues of
xSz(s)
- - a t
the poles of
~ < ~ < ~.
To apply Landau's
theorem we show first that the series
Z(s)
140
Z(s) = ~(2s)
~
la(n) 12n -s
n=l
co
= [ c(n)n -s n=l
satisfies
the c o n d i t i o n s
a2 = -r,
63 = a4 = 0,
of the theorem.
We take
~i = 6i = ½'
T 1 = ½ + r, T2 = ½ - r, T3 = T 4 = ½
= Y1 +
i < i < 4, 61 = r,
and o b s e r v e
that
"'" + Y4 - ~ i - "'" - ~4
=2>½.
By T h e o r e m (2.8)
we have
C(x) =
~ c(n) n<x
la(~) I 2 p>
2
< x
~
la(P) I2
0(~)
= 0(x).
½
=
~ I)< x ½
Therefore
by partial
summation
we observe
Z(s)
=
that
oo ~ c ( n ) n -s n=l
is a b s o l u t e l y Eisenstein function
convergent
series we know
for that
Re(s)
> 1
and h e n c e
( [ 1 7 ], p. 43)
~ = i.
at a p o l e of
F r o m the t h e o r y of &(2s-l) A(2s)
the '
t41
A(2s)E(z,s) A(2s-l)
is holomorphic.
Furthermore
each fixed strip
A(2s)E(z,s)
o I < O < ~2"
Hence condition
The well known behavior of the fact that for
Itl
sufficiently
A(2s)E(s,z)
has at most a finite number of poles in (III) is satisfied.
F-function on vertical
strips and the
large
= A(2s)y s + A(2s-l)y l-s + 0(e -cy)
imply the estimate
Z(s) = 0(eYltl),
as required by condition
Condition by theorem
(VI)
(V).
is simply satisfied because
e(n) = c(n)/~2n-
and hence
(2. 8)
e(n)n = 0(x). n<x
Now where
3 K = ~
and
the only pole of
Z(s)
in the strip
~5 -< ~ j 1
the residue of
Residue s=l
Landau's
xSR(s) s
3 ~2
(f,f > F(½+r)F(½-r)
theorem then gives the estimate 3
c(n) = Co(~)x + O(x5). n<x
X.
is at
s = 1
142
Let us now recall that
la(n)
12n-s
=
n=l
~ c(n)n -s ~ p(m)m -2s n=l
m=l
and therefore
la(n) l2 =
~
c(k)p(h);
kh2=n we then have
la(n>l 2 = n(h)
~ c(k) k 0
A(n),
n<x
where the term in the sum with
n = x
is
is to be weighted by
½
when
x
is an
170
integer.
We shall need the following well known result
Lemma 4.1.
Let
6(y)
be the function of
0 6(y) = {% 1
if if if
0 1
y
([ 6 ], p. 109).
defined by
1
and let
c+iT 1 s ds I o (y,T) = 2--~ f Y "--" s c-iT
Then, for
y > O, c > O, T > O,
IIo(Y'T)-6(Y)I
c < {Y min(l,T-iIlog cT -I
From this lemma we easily obtain, with
y]-l)
if if
y # 1 y = I.
c = 1 + (log x) -I,
co
(i)
]V(x,~)-I(x,T)l
< ~ IA(n)](X)Cmin"
# 0.
We
We divide the
- U < o < -½, t = T.
that the n u m b e r of zeros
and the
and
the first c o n s i s t i n g of the segment
and the second consisting of the segment
the critical strip w i t h
in
llmpl ~ T
along each individual path.
into two parts,
the first piece w e recall from
with
-½< To
O = ~ + iT
O< treat in
c,
178
Therefore,
by slightly
for any zero with
increasing
T
T- 1 < y < T + 1
this were not possible, w o u l d be m u c h larger
if necessary,
the number of zeros
for
Let us recall
~
(s,z) for
from
§3.3
s
P = B + iy of
~.
with
E,
(s,~) = 0(log(It I +
Now we observe
and
2 + it
llll +
IX21) +
that
1 Is-{3
and summation by parts, these terms contribute w e have
and subtract
t
~'
i
derivative
}
of the
§3.3
1
have that for large
E P
If
_ log w - B(~) - [{~_in + ~}; 1
I~'(2+it,~)I
also for the terms in the sum
A.
the expression
w h e r e w e have used well known bounds F-function.
constant
Before proceeding
p
it at s = ~ + it
IT-y1 > A~I
T- 1 < y < T + 1
along the chosen rail
!' (s,~) : ½ yF' (s~ ~i) + ½ ~r'(s-~2) 7--
we evaluate
that
and with an absolute positive
than a constant multiple
must find an estimate
we may assume
and the implied
2,
177
l~'(q+iT,~)l
`i) j ½
always.
~(s,~X~),
Langlands has shown that the stronger result
would follow from the analytic properties of the Euler products using the simple identity, valid for
~
shows, that
Re(>`i) = 0
L(s,~,p).
Now,
not an integer,
~, x %-2n = x___ >` _ x___ % log ( l _ x - 2 ) n=O%--~-n--n X 2
+ X%n~l (~n'X-2n'%
we have
>`~
where
Z
x_ >, ` ( O ) { l o g
(1 + x - 2 ) + max (j>`ll +
I~1 I-l, 1>'21 + 1~2 l-l)},
z(~) x
%(o) = max (Re(>`l),Re(>`2)). Observe that if
complementary series then
~
does not belong to the
%(o) < 0.
We now put together our main result in the following statement.
Theorem 4.2. representation of
(Second yon Mangoldt Formula).
GL2~~ )
llg]l be the norm of
~(x,~)
g
of conductor
f(~)
introduced in §3.3.
= - (log x + ½ ¢(2)(0,~))
¢(1)(0,~)
where the sum
E P
Let
~
be an automorphic
had infinity type If
{%1,>`2}; let
x > 1 and T > i, then
xp + s(~) _ ~ "6-
+ R(x,T),
p
runs over the nontrivial zeros of
~(s,~)
with
llm(p) l ~ T,
180
_i
R(x,T) 11
+
+
IX21)))2T ~-,
and
o~
xkl_2n
( oo) : n:O Oh--7
xk2-2n
+
1
co
H(S,~)
=
~ A(n)n -s, n=l
A(n) >__ 0
as can easily be seen from
-hE
log H(s,~)
[
=
[ ]i+~ C0"
if it exists;
series
co
H(s,~)
in particular
=
Z A(n) n-s, n=l
holds
for
O > O0;
-~ < o0 ~ i.
By
186
log IH(o,~) I = Re log H(O,~)
= log H(g,~)
=
for
g > g0"
But then hence
IH(~)I ~ 1
H(O0,~)
= 0;
g0
H(s,Z)
must vanish to the left of
for
oo ~ a(n)n -g _> 0 n=l
o > gO
does not exist.
contrary
to the fact that
But this contradicts
Re(s) = 1
the fact that
at an infinite number of points on
the negative real axis in order to offset the poles present in the appears in the functional that
L(I,~) # 0.
automorphic
equation relating
To get the non-vanishing
representation
~ ®wit,
where
H(s) of
with
H(l-s).
L(l+it,~),
~it = ~0
and P
character
r-factor that This then proves
we replace ~
To derive a zero free region we use the well known inequality
(i)
to obtain for
{3~H ( g , ~ ) + 4 ~
s = ~ + it, ~ > 1
(g+it,z) +~'(o+2it,z)}
~ II+%(P)n+~(P)nI2(Iog PlP
i3+4 cos (tlog p) +cos(2t log p)}
p~f(g) n=l
> O.
Let for
t > 0
e = e(t) = Ii~II2 log(f(~)(t+
and recall from
§3
the representation
1~ll+
by the
is the grossenP
~ (x) = Ixl it P P "
3 + 4 cos e + cos 28 > 0
~
I~21)),
187
(2)
-
where we now assume that to representations
~
(s,IT) = - Z ~ P
~(s,~)
+ 0(%),
does not contain the local factors corresponding
which are not class one.
To see why this is so observe
P that for
d > 1
l~slog ]~ (l-a(p)p-Sl plf(~) < --
1
Z A(n) n<x
= -
with
[
xp
-
+ s(~
then
Here we simply consider
the Euler product
H(s) = L(s,n X ~ ) L ( s , g X ~ ' ) L ( s , ~
GL2~ ~ )
) + R(x,T)
and
193
i R(x,T)
l-6(ll~l121og(f(~)(l+llll+IX21))) -I.
The elementary estimate
A(n) = ~ X(gp) log p + n r
the minimal
set
basis o f o r d e r
(Erd~s
of order
basis
superset
h,
and
then
of order
]3 is a maximal
proper
Z that are r - m i n i m a l in Z if ISI < r
Nathanson
[3]).
The
bases.
is not a basis
h . The
of order
is a n a s y m p t o t i e
is not an asymptotic
nonbasis hB'
B\S
precisely
B~I>4.
h . If
then
B
h,
B.
is a nonbasis
then
nonbasis
B' ~
B
B
is an asymptotic
of order
Similarly,
of
h
B
if hB/
~,
is a maximal
t
asymptotic for B
nonbasis
every
proper
superset
is a maximal
infinite
of order
h B' ~
asymptotic
sequence
if
n ~hB
B
and
nonbasis
of numbers
for infinitely
all sufficiently
of order
not belonging
Z,
to
n,
large
and
ZB,
many n.
but
n ¢ hB'
In particular,
if
if n I < n 2 < n 3 < ...
then
n. - a ~ B
is the
for every
i
nonnegative
integer
a ~ B
and
all sufficiently
large
n..
Nathanson
[i0] introduced
i
this idea
of maximality,
nonbasis
of order
h
for every
h > Z
which
was
the union
asked
several
ErdBs
proved every
a class
thatif
B~]N
and
Z.
This
tic nonbasis
implies
also
in the sense
that,
order
ISI
_ r
if
nonba
of density
is an asymptotic is contained
that if
B
has
d~j(ZB}
density
< l,
then
each
of
Finally, been
and
of arithmetic
B
he
answered
Turj~nyi.
examples
of maximal
progressions,
and
nonbasis
of order
also Z
for
in a maximal
asymptotic
d(B)
if the surnset
B
= 0
and
is a subset
nonbasis ZB
of a maximal
asympto-
Z. exist asymptotic
if S~]N\B, r,
but
nonbases
then B~
~131 proved
zero, of order
is best
subsequently
I-le constructed h,
progressions.
"nontrivial"
then
h.
of order
Nathanson,
that every
S
B
BUS
becomes
of order
Z
that are
is still an asymptotic an asymptotic
basis
r-maximal
nonbasis of order
of Z
if
T h e l - m a x i m a l nonbases are precisely the
s es.
Turj~nyi
result
F
(Erd}Js and Nathanson [3]).
maximal
nonbases
B~
density
of order
There
not unions
if
F ~ ,
asymptotic
that have
Hennefeld,
showed
of order
nonbases
of aritl~netic
[Z] constructed
that were
and
nonbasis
asymptotic
nonbases
of ErdBs,
Nathanson
finite subset
upper
about
nonbases,
of a maximal
of maximal
results and
the maximal
of a finite number
nonbases
of order has
is a subset
questions
by the following
asymptotic
classified
and
Nathanson Z
possible.
whose
that there
exist maximal
[ii] constructed counting
functions
asymptotic
a class have
of "thin"
order
nonbases maximal
of magnitude
of order asymptotic ~-.
This
2
219
PIennefeld [9] constructed the first e x a m p l e of an asymptotic nonbasis of order
Z that cannot be e m b e d d e d in a m a x i m a l
asymptotic nonbasis of order
Z.
E r d ~ s and N a t h a n s o n [5] constructed a class of "thin" asymptotic nonbases of order
Z such that each set in this class had
O(~x--)
elements not exceeding
and such that no set in this class is contained in a m a x i m a l order
Z.
asymptotic nonbasis of
E r d B s and N a t h a n s o n [6] proved that there does not exist a m a x i m a l
asymptotic nonbasis of order
Z consisting only of square-free n u m b e r s ,
that there does exist an asymptotic nonbasis such that B ~ J {q} number
x,
B
of order
is an asymptotic basis of order
but
Z of square-free n u m b e r s
Z for every square-free
q ~ B. Finally, it is possible to partition ~
that A
into two disjoint sets A
is a m i n i m a l asymptotic basis of order
nonbasis of order "random"
Z; m o r e o v e r ,
elements are m o v e d
Z and
B
and
is a m a x i m a l
B
such
asymptotic
this partition can be constructed so that, as
from
A
to B
to A
to B ...,
f r o m basis to nonbasis to basis to nonbasis . . . and the set eously f r o m nonbasis to basis to nonbasis to basis...
B
the set A
oscillates
oscillates simultan-
(Erd~s and N a t h a n s o n [4]).
It is not k n o w n w h i c h of the results above are true for bases and nonbases of orders
h > 3.
In this paper I consider a combinatorial analog of m i n i m a l bases and m a x i m a l nonbases.
Let
,~(~)
denote the collection of all finite subsets of ~ ,
~ ,~(~) . Denote by
h~
distinct sets belonging to order
h . Otherwise,
but finitely m a n y order then
h. ~
~
the collection of all unions of h
~.
If h ~
=~(]N),
~
is a union nonbasis of order
elements of Z there exist asymptotic union nonbases of order
e m b e d d e d in m a x i m a l
asymptotic union nonbases of order
k n o w n if there exists a m a x i m a l
h.
h
Similarly,
that cannot be
Indeed, it is not
asymptotic union nonbasis of order
h for any
h>2. Notation.
T h e natural n u m b e r s
case letters denote natural n u m b e r s natural n u m b e r s .
iN are the nonnegative integers.
and capital R o m a n
letters denote sets of
Capital script letters denote sets of sets of natural n u m b e r s .
interval of integers
a < n
Z,
h.
Let h >
Z,
Clearly,
however,
But it is not true
h > Z contains a m i n i m a l asymptotic ~
such that
Z for every finite subset ~
is an asymptotic union nonbasis of order
THEOREM
For
Indeed, I shall construct a basis
is an asymptotic union basis of order ~\~
S ~
~,
~\
2
but
Z for every infinite subset
and let TI, T z , . . . , T h
be a partition of IN into
n o n e m p t y sets at least two of which are infinite. T h e n ~_jh i=l h.
£/(Ti)\{ ~ } )
is
a m i n i m a l asymptotic union basis of order
Proof.
Let ~
for j = l,...,k,
=~_Jhi=l ~ ( T i ) \ { ¢ } )"
then
X(~Tij ~ ~
Let X ~ - ~ ( ] N ) , X / ¢ .
for j =i ..... k,
If X ( - ~ T i . / ¢ J
and
k
X = k_j ( X ~ T i . ) ~ k ~ j=l
hence h ~
=,~(Eq)~{~}
•
~h~
3
M o r e o v e r , if X ( ~ T . / ~ f o r each i, i
then X =~jh
i=l
(Xf'-~Ti) is the unique r e p r e s e n t a t i o n of X as the union of h e l e m e n t s of ~ Let
S ~ g
,
say,
S ~ ~(TI)\{~
} . At least two of the sets
.
T.l are infinite,
hence ~jh T. is infinite, and so there are infinitely m a n y sets X ~ "~r(]N) such i=Z i that X~-%TI = S and X~-~T.~ l/ for all i = Z, 3 ..... h. But X / h ( S \ { S } ) , and so
~kk{S}
is an asymptotic union nonbasis of order
E a c h m i n i m a l asymptotic union basis
~
h.
This proves the T h e o r e m .
constructed above has the
property that if B ~ (~ , then every n o n e m p t y subset of B the "trivial" m i n i m a l asymptotic union bases.
is in ~
T h e following L e m m a
.
T h e s e are
will be
applied to construct a class of nontrivial m i n i m a l asymptotic union bases of order Z,
and also to construct union bases of order
asymptotic union bases of order
LEMMA.
Z that do not contain any m i n i m a l
Z.
Let Rk be a n o n e m p t y s u b s e t of
T h e n there is a family
~k+l
[1, nk] ,
and let n~+ l >__nk + 3.
of subsets of [i, nl~+l] with the following properties:
224 (i)
[nk + I, nk+l] ( /$k+l'
and
B(-~[nk + I, ink+l] / 6
(ii) If X C [I, nk+1] and Xf-h~[nk + I, n]I"
{Rk});
if R' ~
It,
Jl ~ ~ k + l
X : (1%1U S' U I i ) U
and 1%Z u
If I{l = It'
then X : (It' <J J ) U (S' U I)
. Suppose that X I / I. Then
IR_kl > I, we canwrite
1%1v S' Y
S' <J I ( (~k+l"
X I : Jl u Jz for
It' : 1%1[.J 1%Z' where
Itl~ i%
JZ C (~k+l' hence
( i t z U JZ ) : 1%' U S' ~ X I
This proves (ii). Finally, let 1%U I = B I U
B Z for s o m e
B.(~I/If
for i = 1, Z, then Bi(-~I~ ~
for s o m e
I, say,
and
B1 = L
{ ~ } . But I ~ Z7 .
i : i, and so B l : S' Y L
It follows that i~ c B Z.
B Z ( ,~([i, nk]), and
n.k]). If
Therefore,
B.~-~I I i =
But (itU l)(-~S : ~ , and so S l :
But this is impossible if B Z ¢ ~k+l'
hence
B e : 1%. This proves (iii).
Case II: Suppose sets of the f o r m
V
BI, B Z ~ ~ k + i U J ( [ l ,
I%1
: 1, say,
S' Y I , where
%
: { r }.
Let ~k+l'
S' _~S = [i, nk]X{r}
consist of all
and I' _~I, I' / ~.
225
Clearly,
~k+l satisfies (i).
Let X C [1, nk+l] with X(~I # ~ . If r ~ X, then
X ( 6 k + i C Z ( ~ k + l ~ J {P~k} ). If r e X, then X~{r} ~ O~k+ 1 and X : (X~{r})U { r]. ¢ Z(~k+IU {R.k}). Thus, ~ k + I satisfies (ii). Finally, if P~kU I : B1U BZ, where
BI, Bz ( O~k+iU~([l, nk]), then r e B.I for some
BI ~ ~k+l'
hence
BZ e ~k+l
and
B I ~ [i, nk] and so B I = { r} = R-k.
B Z = I. Thus, ~ k + l
i, say, i = I. Then
Then
I C BZ,
hence
satisfies (iii). This completes the proof of
the L e m m a . THEOREM
4. There exist nontrivial minimal asymptotic union bases of
order Z. THEOREM that, if # C
5. There exist asymptotic union bases
8,
then
~\~
~
of order Z such
is an asymptotic union basis of order Z whenever
I ~ I < o0, but an asymptotic unionnonbasis of order Z whenever particular,
~
In
does not contain a minimal asymptotic union basis of order Z.
Proofs. Let {nk} for all k > l .
I~[ =~.
be a sequence of positive integers such that nk+ I > ~k + 3
I first construct inductively a sequence of sets ~ k ~ ( [ l ,
nk])\{ ~ }
and sets P~k ~ U k ~i" Let ~i =L~([I, nl])\{ ~ ] Suppose that ~l'" ~ k and i=l . . . . i~i,. . . , Rk_ 1 have been determined. Choose any Rk ~ uki=l ~i" Let ~k+l ~-([I, nk+l])\~([l, nk] ) satisfy conditions (i)-(iv)of the L e m m a . o0 UM=I @k"
Clearly,
~ ~ Z~
then X ¢ ~ i : Z~l--~Z(~" some unique k > l . Z ~ =~(~)'N{~ },
since ~ { ~ . Let X ~ ~-~(]m), X / ~ . If x C
Otherwise,
~
X E Z(~kll~{Rk})~- Z~.
for
Thus,
is an asymptotic union basis of order Z.
Let B_kU Ink + 1, nk+l] : BlkJ B2, where
B1, B Z ~ 03 . Since R k (
uki=l ~3iC~([l'-- nk]), it follows that BI, BzC[I,_ %+i]. implies that BI, BZ ( <jk+li=l~k+l C_ ~k+l U ~([l, nk]). Zemma,
[i, nl] ,
XC[I, nk+l] and X~[nk+l, nk+l] / 6
By condition (ii) of the g e m m a , and so
Let ~ :
either B 1 : P~k or B Z : P~k" Thus,
Condition (i) of the L e m m a By condition (iii) of the
P~kU [nk + I, nk+l] ~ Z(~\{P~k}).
The sets P~k~J [nk + i, nk+l] are pairwise distinct, although the sets R k themselves need not be distinct. H o w shall we choose the sets P~k ? infinitely often as an Rk; that is, if B ~ ~ Then
RkU
Suppose that every set B ~ ~
, then B = P~k for infinitely m a n y
[nk + l, nk+l] ~ g(~\{ 13} ) for infinitely m a n y
minimal asymptotic union basis of order Z. Since IB I > i, condition (iv) of the L e m m a T h e o r e m 4.
is chosen
implies that
k,
k.
and so 03 is a
03 contains sets B with ~3 is nontrivial. This proves
226 Now
~C
suppose that every set B ~ ~
~ " If R.k ~ ~,
asymptotic
union
then
nonbasis
w e have [_JOOk=t~ k C- ~ \ ~ 2
is chosen exactly once as an Rk.
P,_k~J [n k + i, nk+l] j Z(~), o~ order
and
Z whenever
P,-k ( 0 ~ \ ~
and so ~\~
r"fl =~"
for all k >t._
X(-h[n k + I, nk+[] / ~ . By condition (ii) of the L e m m a , if k > t , ~\~
and so
Z(~\~ 0) contains all but finitely m a n y
is an asymptotic union basis of order
Theorem
5.
But if
l~r
Let
is an t. H S _ C ~ ,
with
sets X
h,
with
then T h e o r e m
6
Ixl < t. ]But
is an asymptotic union basis of order
h.
proves the Corollary. ~t = {Xc~(Eq)
Ilxl > t}
is an asymptotic union
h that is not contained in any m a x i m a l asymptotic union nonbasis
7.
Let h _> Z, and let ~
be a m a x i m a l asymptotic union
h that is also an asymptotic union nonbasis of order
Zh - Z.
non-
If
then S(-~T : ~ .
Proof.
By m a = ~ m a l i ~ , both
bases of order h. Xc
But the
h.
THEOREM
and
Therefore,
Ixl = k.
forall X ~ ( ~ I
if IXl > t. Therefore,
In particular, the set
of order
ha
Then
contains all but finitely m a n y
This contradiction
IX[ = k.
Therefore,
m a x i m a l asymptotic union nonbasis of order
a
h and
be an asymptotic union nonbasis of order
contains all sufficiently large sets.
Proof.
is S.
h,
IxI = k.
and so 0"5 (,9 {S } is an asymptotic union basis.
contains all but finitely m a n y
only k- element setin h(0~[.J { S } ) t h a t is not in h03
that h @
with
is a m a x i m a l asymptotic union nonbasis of order
, then S ~ 03
h(0"~J {S})
~3 is a m a x i m a l asymptotic union nonbasis of order
~ U { S} and 45 U { T}
Therefore, there i s a s e t
h((~L._){T}),
but X {
(Zh-Z)(~.
X /S,T Then
are asymptotic unio~
such that X~ h ( ~ U {S})
X{hO~,
and so
x : sU BzU ... U Bh : TU B~U ... U B~,
228
where
B., B! ¢ ~ i
and so
for
X = BzU
. . ~J. B h.U . B.~ U
COI~OLLAI~Y.
S,T¢ ~ ,
i = Z ..... h.
If S~-~T = ~ , then
TC
BzU
... V
Bh,
1
then
Let
~3
s~T/¢.
U
B hI ~ (Zh-Z)~
be a m a x i m a l
But this is a contradiction.
asymptotic union nonbasis of order
Z.
229
5. i.
Let
~
~(~)\~.
Open problems
be an asymptotic union nonbasis of order
Then
~
is r - m a x i m a l if ~ L_J ~
h,
and let
is an asymptotic union nonbasis
of order
h whenever
I~I < r,
but ~ ~ _ J ~
of order
h whenever
i#I > r.
T h e 1 - m a x i m a l asymptotic union nonbases are
precisely the m a x i m a l
asymptotic union nonbases.
asymptotic union nonbases of order case
r = 1 and 2.
Then
~
Let
b e c o m e s an asymptotic union basis
h?
D o there exist r - m a x i m a l
This is not k n o w n even in the simplest
h = 2. (~
be an asymptotic union basis of order
is r m i n i m a l
h,
and let ~ _ ~
if ~3\~f is an asymptotic union basis of order
.
h whenever
I.#l < r,
but ~ \ 2
I~I > r.
T h e l-minimal asymptotic union bases are precisely the m i n i m a l asympto-
tic union bases. all r > Z and order
b e c o m e s an asymptotic union nonbasis of order
~
h whenever
D o there exist r - m i n i m a l asymptotic union bases of order h > 2?
h for
A r e there nontrivial m i n i m a l asymptotic union bases of
h > 3? 3.
Classify the m i n i m a l asymptotic union bases and m a x i m a l
union nonbases.
asymptotic
A r e there general criteria that imply that an asymptotic union
basis contains a m i n i m a l asymptotic union basis or that an asymptotic union nonbasis is contained in a m a x i m a l 4. that ~
asymptotic union nonbasis ?
Is there a partition of .~(]N)
into two disjoint sets
is a m i n i m a l asymptotic union basis of order
asymptotic union nonbasis of order 5.
Z and
~ 03
and
6~
such
is a m a x i m a l
2?
If w e consider intersections of sets instead of unions of sets, then w e
find a n e w series of u n a n s w e r e d combinatorial p r o b l e m s about define an asymptotic intersection basis of order such that all but finitely m a n y
sets in
~(l~)
of h not necessarily distinct sets in ~ bases exist?
h
.~(]N). F o r example,
for ,.~'(]N) to be a set ~
~(]N)
can be represented as the intersection
. D o m i n i m a l asymptotic intersection
D o e s every asymptotic intersection basis for
~(l'q)
contain a
m i n i m a l asymptotic intersection basis ? 6. Then
B
Let
Q
be the set of square-free positive integers, and let B ~ Q .
is an asymptotic
LCM
basis of order
h
for Q
if all but finitely m a n y
square-free integers can be represented as the least c o m m o n of B.
Similarly,
B
is an asymptotic
GCD
basis of order
multiple of h h for Q
elements
if all
sufficiently large square-free integers can be represent.ed as the greatest c o m m o n
230
divisor
of h
elements
of
B.
We define
LCM
and
GCD
bases,
nonbases,
and asymptotic nonbases similarly.
Combinatorial t h e o r e m s about union and
intersection bases and nonbases for
._~(~xI) are equivalent to multiplicative t h e o r e m s
about
LCM
and
GCD
Z = P0 < Pl < PZ < "'" q : c~(]N) -~ Q LCM
by
bases and nonbases for Q
be the sequence of p r i m e s in ascending order.
q(B) = IIb~B qb
[q(B l)..... q(Bh) ] and
follows that ~ ~.~(IN) nonbasis) of order asymptotic
LCM
h
in the following way.
for all B ¢ ._~(]N). T h e n
q(Bl(-~... (-~B h) -- G C D
Let
Define
q(BiKJ ... [-J B h) =
(q(B I)..... q(Bh) ). It
is an asymptotic union (resp. intersection) basis (resp. for
.~(]N) if and only if q((~) : {q(B)[B ¢ (~} _ C Q
(resp.
GCD
) basis (resp. nonbasis)of order
Thus, combinatorial t h e o r e m s for
.~(]N)
is an
h for Q.
can be translated into multiplicative
t h e o r e m s for Q. It is natural to consider set of al_~lpositive integers. asymptotic
LCM
(resp.
elements of B.
W e define
nonbases similarly. plicative n u m b e r
T h e set GCD
integer is the least c o m m o n
LCM
and B
GCD
bases and nonbases for the
of positive integers will be called an
) basis of order
h if every sufficiently large
multiple (resp. greatest c o m m o n LCM
and
GCD
divisor) of h
bases, nonbases,
and asymptotic
This generates a n e w series of unsolved p r o b l e m s in multi-
theory.
T h e s e can be translated into combinatorial p r o b l e m s
about union and intersection bases for multisets. Graham, all n u m b e r s
Lenstra, and Stewart
of the f o r m
[7] have observed that the set consisting of
Z • 3 n, n = 0,1, Z, 3 .... , is a m a x i m a l
nonbasis for the positive integers.
T h e existence of a m a x i m a l
asymptotic
LCM
asymptotic
nonbasis for the square-free integers is still an open problem. Finally, there is an analogous series of p r o b l e m s about m i n i m a l bases and m a x i m a l
nonbases for the positive integers under ordinary multiplication.
231
References i. M.
D e z a and P. ErdBs,
Extension de quelques t h e o r e m e s
densities de series d I elements de
N
sur les
a des series de sous-ensembles
finis de
N,
Discrete Math. 1_Z(1975), 295-308. 2. Amer.
P. ErdBs and M.
B. Nathanson,
Maximal
asymptotic nonbases,
Proc.
Math. Soc. 4__~8(1975), 57-60. 3.
numbers,
P. ErdBs and M. Proc. A m e r .
4.
B. Nathanson,
Math.
P. ErdBs and M.
Oscillations of bases for the natural
Soc. 53(1975),
]3. Nathanson,
infinitely oscillating bases and nonbases,
253-258.
Partitions of the natural n u m b e r s
Comment.
into
Math. Helvet. 5__~I(1976), 171-
18Z. 5.
P. ErdBs and M.
in m a x i m a l 6.
nonbases,
B. Nathanson,
N o n b a s e s of density zero not contained
J. L o n d o n Math. Soc. 15(1977).
P. ErdBs and M.
B.
Nathanson,
Bases and nonbases of square-free
integers, preprint. 7.
It. L. G r a h a m ,
H. W.
Lenstra,
Jr., and C. L. Stewart, personal
communication. 8. Angew.
E. H~rtter, Ein Beitrag zur Theorie der Minimalbasen,
J. Reine
Math. 19___~6(1956),170-Z04. 9.
J. Hennefeld, Asymptotic nonbases not contained in m a x i m a l
asymptotic
nonbas es, preprint. I0. number
M.
B. Nathanson,
theory, J. N u m b e r ii. M.
M i n i m a l bases and m a x i m a l
Theory_6(1974),
B. Nathanson,
s-maximal
nonbases in additive
324-333.
nonbases of density zero, J. L o n d o n Math.
Soc. 15(1977), 29-34. IZ.
A. StBhr, GelBste und ungelBste F r a g e n ~iber B a s e n der n~turlichen
Zahlenreihe, 13. Number
J. l~eine A n g e w .
S. Turj~nyi,
Theory9(1977),
Math.
On maximal 271-275.
194(1955), 40-65, 111-140. asymptotic nonbases of density zero, J.
REMARKS ON MULTIPLICATIVE
Institute
My principal
Atle Selberg for Advanced Study, Princeton~
We begin by recalling
functions
08540
New Jersey
reason for choosing this rather elementary
attention to the uses of multiplieative
io
FUNCTIONS
topic is to draw
in more than one variable.
the standard definition
of a multiplicative
function of one variable defined on the positive integers:
it is a function satis-
fying the conditions
(i.i)
f(m) f(n) = f(mn)
for
(re,n) = i,
and
(1.2)
f(1) = i.
I have never been very satisfied with this definition~ define a multiplicative
and would prefer to
function as follows:
Write
(1.3)
n = ~
pa, P
where the product extends over all primes (so that all but a finite number of the a
are zero).
Let there be defined
negative integers
such that
(1.4)
f (0) = 1 P
p
a function
f (a) P
on the non-
except for at most finitely many
p.
Then
f(n) = ~ - ~ fp(a) P
defines a multiplicative This definition f(n)
for each
singular if
f(1) = i~
function.
is clearly more general than the previous
f(1) = 0,
we say that
f(n)
otherwise we call is normal.
*)It should be noted that it permits
f(n)
f(n)
regular.
one *).
If finally
The class of multiplicative
to vanish identically.
We call
functions
233
defined by the standard definition coincides with the class of normal multiplicative functions according to our new definition° With the new definition it remains true for instance that if are multiplicative,
~
f(d)g(~l
\47
dln it also remains true that if f((a,n))
f([a,n]) *)
and
g(n)
then so is the convolution
f * g(n)=
then
f(n)
f(n)
,
is multiplicative and
is multiplicative.
a
a positive integer
However, with our new definition,
are also multiplicative,
f(an)
and
something which is not necessarily true with
the standard definition° Another advantage is that the new definition can be used without change to define multiplicative functions of several variables. If we denote by
[n}r
an
(1.5)
r-tuple of positive integers
nl,...,n r
and write
In} r = I I P ~a}r P
to denote that ai n.i = I I p P we say that a function
for
i = 1,2,...~r,
f(nl,...,n r) = f([n}r)
is multiplieative if we can write
it in the form
(1.6)
f([n]r ) = ~-~ fp({a]r), P
where the functions For each integers,
p,
if
f({n}r )
satisfy the following conditions.
fp(al,...,ar)
fp(0,...,0) = i
Again, writing that
f ([a}r) P
is defined on the
r-tuples of nonnegative
except for at most finitely many
[l}r
is singular if
for the
p.
r-tuple all of whose entries are i, we say
f({l]r) = O,
regular if
f([l]r) ~ O,
and normal
f([l}r) = i. It is easily seen that if one keeps some of the variables fixed in a multi-
*)We use
[a,n]
to denote the least common multiple of
a
and
n.
234
plicative function one gets a function which is multiplicative in the remaining variables. Let us finally mention that in case of functions of one variable the class of multiplicative functions defined by (1o4) could also be defined by the requirements:
(1.7)
f(m) f(n) = f([m,n]) f((m,n))
for all positive integers
m
and
n.
This is, in spite of its simplicity~ not as
practical as the constructive definition (1.4).
Also one meets complications when
trying to adapt it to the case of several variables°
2.
We shall now concentrate on functions of two positive integral variables~
though as of yet we shall not necessarily assume them to be multiplicative. that a function
f(m~n)
is symmetric if
f(m~n) = 0
n > m~
and finally normal lower triangular if
all
for
f(m,n) = f(n~m)~
We say
lower triangular if f(n~n) = i
for
n. If
t(m,n)
is normal lower triangular and we have two sequences
xm
and
connected by the relations
(2.1)
x
m
=~
t(m,n) Yn n
then there exists a unique normal lower triangular function
(2.2)
Ym = ~
t*(m~n)
such that
t*(m,n) x . n
n
t
and
t
are
connected
(2.3)
where
by the
relations
~ t(m,~) t*(~,n) = 6m, n,
6
m~ n
is the Kronecker symbol, or, alternatively we have
(2.3')
~-~ t*(m,~) t(~,n) = 6m, n.
If we assume that
t(m~n) nlm.
is multiplicative,
t(m,n) = 0
unless
plicative.
Namely~ let us define
it follows immediately that
It is not hard to see then that ~(pr, pS)
for
r > s
t*(m,n)
is also multi-
by the relations
Ym
235
>2~
(2.4)
t(pr pt) ~(pt,pS) = 6 r,s
s N
and
x I = I.
Writing
f(m,n)
Q
under the side in the form given by
(2o12) we obtain 2
(3~i)
Q(x) = ~
g(~)I~m t ( m , ~ ) X m l .
*)We assume for simplicity in this argument that f(l,l) (and therefore also g(1)) equals I. This is no restriction since we could otherwise divide by f(l,l) which is positive.
237
Writing further
(3.2)
Yn = ~ t ( m , n ) m
so that also
Yn = 0
for
n > N,
(3.3)
Xm,
we get
Q(x) =
2 ~ g(n)Yn " n i,
a weight
for
Wn ~ O.
d > Z
We assume that
and leaving the other
n
W =~w
%d
with each of which there is n < =o.
Writing
k I = i;
as free real variables,
we form the
expression
(4.1)
Q(%) = ~ w n n
Clearly
Q(%)
~
.
)
is always an upper bound for the sum of the weights
which remain after we have removed those that lie in any of the residue classes modulo each weights
wn
the quadratic
pr. form
of the integers
w(p r)
Under rather general assumptions Q(~)
excluded
about the set of
can be written in the form
Q(X) = QI(X) + R where
Ql(k) = W
and
f(d,d')
and
R
I%dl.
f(d,d') kd kd'
is a symmetric multiplicative
is a remainder The machinery
function (positive definite,
term generally bounded by a simple quadratic
from the previous
subject to the side conditions determined
~ d,d'
by the requirement
on the that
section then applies, %'s,
R
of course)
form in the
one can minimize
the choice of the parameter
Z
QI(~) is then
should be small enough not to spoil the
result. We shall apply this technique (so that we assign the values
~
= i~
introduce a symmetric multiplieative if
r = s
d'
are compatible
now get
or if
rs = O, if
to the case of an interval for
n 6 Ix
function
otherwise we define
E(d,d') = 11
otherwise
and
~(d,d')
I
Wn = 0
x
outside
by defining
E(pr,p s) = O.
of length I x.
X We
E(pr~p s) = 1
We say that
d
they are said to be incompatible°
and We
239
Q(~) =
E nEl x
k
< x E f(d,d') ~d ld' d,d'
)
(4°2)
+ E
IXHI
{~d' [ w(d,d')
~(d,d'),
d~d' where
(4.3)
f(d,d') = w([d~d'I) [d,d']
Here
w(d)
is the multiplicative
~(d,d').
function defined by
w(d) = I I r
00(i) = i
and
W(P r)°
p lld
An alternate form of the upper bound for the interval follows *) . w(u) > I
Consider a function for
u
in
I
--
and
w(u)
can be obtained as
defined on the real line~ such that
w(u) > 0
x
Ix
always.
We furthermore require that its
--
transform
fourier
W(V) = 7 W(U) e 2~iuv du .oo should vanish identically
for
i
Iv I >--~ .
We then have
z ~
(4.4)
Q(%) _ < ~ _ao
w(n)~ % LnX(d)
It can be shown that we can choose w(0) < X + Z 2.
d•2
= wA(O)
w(u)
E d,d'
f(d,d') k d ~d' "
satisfying our conditions and such that
Thus we get
(4.4')
Q(%) ~ (X+Z 2)
~ f(d,d') %d %d'" dgd'
To use the results of the previous section to minimize the quadratic form on the right hand side of (4.4')~ we observe that we have and
f(pr,pr) = f(pr, l ) = f(l,pr) = ~ ( r )
Writing
P (4.5)
we have
8(p r) = 1 -
g(1) = i,
*)See Selberg [i]o
and for
r > 1
~ ~(ps) s l<s s 8(pr-l)-@(p r) *. r s. Jt (p ,p) =
(4.8)
1
@(pr-l)
'
8(pr-l).~(p r) 6(pr-l) '
if
s = 0
if
O<s