Number Theory Day

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
q

c

coroliary.

T h e polynomial 4(x3-x Z) + I

ha s i)

exactly

ii)

exactly

iii)

one linear 3

factor

Two

n

p ~ n ( r o o d 11) w h e n

(~)

provided no linear

factors

( r o o d p)

power

series in x

if

p --- r ( r o o d li)

integers

u

co

x

and

'',, ( I - x n) 1

From

CO Z =Z

c xn I

n

where

r

(~)

r

(~-)

= +l,

= -1 . = +1,

provided

v.

are said to be congruent

of the difference is congruent to 0 (rood Z).

follows

if

linear factors ( m o d p) i f p ~ r ( m o d ll) w h e n Z p = u + llv Z for some integers u and v.

3p = u Z + llv Z for some

§ I.

( m o d p)

( m o d Z) if every coefficient

oo

23

cO

c x

n

=

x

I I (l-xZn)(l-xZZn)(m°d

n

1

Z)

.

(i)

1

Now 3uZ_u cO

oo

l[ (l-xn) = I + 23 (-l)U[ x 1 1 So changing

x to x

Z

+x

3uZ+u Z

]

(2)

Z

I 1

oo 3uZ+u] (l-xZn) = 1 + ~B (-l)U[ x 3uZ-u + x 1

(3)

T h e right side of (3) can be written oo

Ux3U2 (-i)

+u

-00

Thus

II ~ i Changing

x to x

ii

(l-xZn) =23c° (-l)Ux3UZ+U--- ~S°° x3UZ+U -oo -oo

(4) by o0

(i) and

3uZ+u+ll(3vZ+v) X

(rood Z) .

(6)

(6)

(6u+l)Z+ll(6v+l)Z

oO

~ c x n 1 Now

(rood Z) ,

(5) produces

I I (l-xZn)(l-xZZn) - 23 1 u, vc Z From

(4)

in (4), w e obtain oo co ll(3vZ+v) [ I (I-xZZn) =- 23 X 1 -cO

and multiplying

(rood Z) .

(7) m e a n s

n

23 U~ V E Z

the following:

x

iZ

(rood 2) .

(7)

Theorem

h

n

where

is the n u m b e r

d

(8)

(rood Z)

=-d

c

n

of solutions of

n

(6u+l) 2+ii (6v+l) Z n

§ g.

From

Theorem

Theorem

=

(9)

IZ

1 w e shall deduce.

Z: c

( m o d Z)

~0 n

if n 2--Z, 6,7,8,10

Proof:

T h e result will follow f r o m

(rood Ii)

Theorem

1 once it is s h o w n that w h e n

(I0)

( r o o d ll)

n - Z,6,7,8,10

we have

d

= 0

(il)

.

n

Now

by

(9), (ii) follows i m m e d i a t e l y f r o m

(I0), since if d

w e r e not

0

n

(iz)

IZn = (6u+l) Z + ll(6v+l) Z

would h a v e solutions.

Since

IZ ---1 (rood ll) a n d

Ii ---"0 (rood ii), (IZ) w o u l d

i m p l y that n E (6u+l) Z

contradicting § 3.

(I0).

N o t e that in the result of § Z,

Theorem

(iz)

(rood ll)

n w a s not restricted to be a p r i m e ,

Z is stronger than the result stated in the introduction.

n will be a p r i m e

p.

thus

In this section

r

Theorem

3:

If p---r (rood ii) w h e r e

( ~ ) = +I,

then

Z

divides

c

if P

Z p

for s o m e

integers

We

u

and

= u

+ llv z

v.

need a number

of e l e m e n t a r y

lemmas

from

the theory of binary quadratic

fo r m s .

Lemrna

h

If p ~'1,3,4,5,9 Z + llv Z

(rood 11) then either

i) p = u or

ii) p = 3 u Z + Z u v + 4 v Z Z but not both.

Exaraple:

[ Thus

either

p

or

p = 47 = ii(i)Z + (6) Z.

choice of u

and

v

3p

is of the f o r m

We

cannot h a v e

for this w o u l d m e a n igp

=

u

+ llv Z.]

p = 3u Z + Zuv + 4v Z for a n y

that

564

=

(6u+_Zv) 2 + 4 4 v 2 .

It is easily verified that this is not the case. Recalling T h e o r e m

1 for

n = p,

with

p =- 1 , 3 , 4 , 5 , 9

(rood II) as in Section

Z we have c

---d P

where

is the n u m b e r

d

(rood Z) P

of solutions of equation (9) for

n = p,

i.e. Of

P

(18)

iZp = (6u+l) Z + ll(6v+l) Z .

From

the v e r y i m p o r t a n t Satz

sentations of a n u m b e r forms

n

Z04

of L a n d a u

by the w h o l e

of a given discriminant, w e

[Z] , giving the n u m b e r

set of primitive

r e d u c e d binary quadratic

see that (18) does in fact h a v e p=u

of repre-

Z

solutions if

Z + fly Z

a n d only one solution if p = 3u Z + Zuv + 4 v Z .

Hence not divide

Cp,

Z

if p = u Z + llv Z a n d the opposite case, i.e., Z P holds iff p = 3u Z + Zuv + 4vZ; w h e n p ='='I,3,4,5,9 (rood II). divides

e

does

4.

It is a special (proved) case of the Weil conjectures that the n u m b e r

of

N P

solutions of the c o n g r u e n c e y

is, for

p } ii,

Z

= 4(x3-x z) + 1 (rood p)

given by

N

= p

P

Clearly

(i) N

-

c

.

(19)

P

is e v e n if 4(x3-x Z) + 1 is irreducible

( m o d p),

while

ii) N P

is odd if 4(x 3 from

(19),

c

Z) + 1 has exactly one or three linear factors

(rood p).

Thus,

is odd in case i) and even in case ii). This leads to the proof of P

the corollary stated in § 0.

Cowles'

proof of the corollary c o m e s

of the D e d e k i n d zeta function of the field K

arising f r o m

Q

f r o m a study

by adjoining a root

of 4(x3-x Z) + 1 = 0.

§6.

In conclusion w e recall a f e w other properties of the coefficients i) p + 1 - c

- 0

c : n

(rood 5)

P ii) c

mn

= c

m

c

n

Relations of i) with s o m e

if (re,n) = i.

identities of R a m a n u j a n

are pointed out in a recent

(unpublished) joint note of J. C o w l e s and S. Chowla.

l~eferences

i. S. C h o w l a and M. x ~ I

2.

Cowles, O n the coefficients

c

n

in the expansion

(l_xn)Z(l_xll n)Z = ~ c x n , J. reine angew. Math. Z9Z(1977), 115-116. 1

n

E. Landau, " E l e m e n t a r y N u m b e r

Theory," Trans. by J. E. G o o d m a n ,

Chelsea,

N e w York, N . Y . , 1966. 3.

G. Shimura, A reciprocity law in non-solvable extensions, J. l~eine Angew. Math. Z21(1966), Z09-ZZ0.

10

Z.

Remarks

on c o n g r u e n c e c

n

c~ 2

n=l

of the coefficients

~ c xn = x n

c

m

c

n

(l _ x n )Z( l _ x l l n )g

n

n=l

It is a striking property of the

These

properties

in the expansion

-- c

c

mn

defined a b o v e that they satisfy

n

if (re, n) = 1 .

,

coefficients a r e also i m p o r t a n t in the study of reciprocity lairs in non-

solvable extensions

of the rationals,

e.g.,

see S h i m u r a ' s

p a p e r [Z] .

Another

striking p r o p e r t y is given by

(A)

c

---p + 1 (rood 5),

for p r i m e

p ~ ii .

P This is implicit in the theory of elliptic curves, with the W e i l conject*~res.

See the p a p e r [3]

There

(A)

is a m p l e

of S h i m u r a ' s

support of

(B)

was

extend

c

0- (n)= E d a. a dl n

in the table for

p < Z000

formulas

unsuccessful.

n

(A)

by a s i m p l e a r g u m e n t

---~ (n) (rood 5), 1

When

a = I,

for

to

(n, ll) = 1

w e will write

o(n)

in place of o l(n).

But w e h a v e results of interest w h i c h

recursive m e t h o d ,

late the coefficients consequence by A.

proof of

of l ~ a m a n u j a n w o u l d aid us to do this.

point of v i e w m i g h t lead to an e l e m e n t a r y a n obvious

Mathematicae.

by Trotter at the end

our original p u r p o s e to give a direct but e l e m e n t a r y

that s o m e

§1.

by Tare in fnventionaes

p a p e r [Z] .

In this p a p e r w e

where

recently studied in connection

Cn.

(A).

m e n t i o n h e r e the estimate

w o r t h recording.

thought

Our

O u r p a p e r also includes to calcu-

[ cp I < Z V'-~ w h i c h is a

hypothesis for plane c u r v e s over finite fields, p r o v e d

Weil.

A recursive f o r m u l a for the Proceed

We

h a v e so far b e e n

apparently not r e c o r d e d in the literature,

We

of the R / e m a n n

proof of

seem

We

(A).

It

by "logarithmic

c . n differentiation"

of the equation

11

oo

E

oo n

n=l

log

(Ecnxn)

= log

x+

n=l

ZE log(1-x

Enc

n

n) + ZE log(1-xlln),

x

n-1 1 -

Ec

n (l - x n )z(1 - x lln )z

c xn = x

n

n

M u l t i p l y the last relation b y

x

x

x

nx -

2

E

-

n=l

-

l-x

n

and multiply by

lln- I nx ZZE - 1 - x lln

-

Ec

x

n

n n

Enc

Then,

n

x n = (Ecnxn

)(1-ZE

nx

lln

- 22E

l-x

n

nx ----~n l-x

)"

since co n co E - n -x = E o- ( n ) x n n=l 1-x n n=l

and co E n=l

~nc

x

n

= (~c

n

lln

co E 0(1%)x lln,

-

nx lln

l-x

xlln

x )(l-ZZ~(n)xn-zz~(n) I% n

n=l

).

Thus

comparing

coefficients of

x

n

o n b o t h sides:

nc

n

= c

n

- Z

n-1 E t=l

c 0-(n-t) - ZZ t

c o-(v) .

Z

u

u+llv=n

u, v > 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
C n ~ n =CI. In the case of discriminant sition g r o u p

G,

-2700,

we

can define a character of the c o m p o -

by 3 X ( C 7 C 3) = e

tl

6 e

t2

31 where

0 <_ tI < 3, 0 i tz < 6, F o r each

and e a c h choice of tl, tz gives a different character.

tl, tz w e h a v e b e e n able to d e t e r m i n e the "twist" 18

T ( s ) = ~ x ( C .J) Z ( s , Cj) j=l by c o m p a r i n g

T(s)

with triple products of the type:

H

2k

12

1

l-

=I x(mod k) p

pS

where 1

X p, k(a) =

if

X

if

is just a n ordinary Dirichlet character

extended over all rational p r i m e s L - s e r i e s of K u m m e r

fields.

soluble

z~__i e

and

x k - a(p)

p.

(rood k),

and the inner product is

T h e s e are nothing m o r e

[See D. M .

abelian L-functions," Israeli J. M a t h . ,

Goldfeld " A large sieve for a class of non1973.]

W e find that

10

Z(s,C I) + Z(s,C Z) + Z

Z(s, Cj) = Z{K(S) j=3

where

K : O(J:~),

Z(s, CI) + Z(S, Cz) + ZZ(s,C3) + 2Z(s, C4) - Z(s,C

5)-

- Z(s,C

7) - Z ( s , C

Z(s,C

6) 8)

- Z ( s , C 9) - Z ( s , Cl0)

{K(S) ~(s) where

K = Q(~/5),

than A r t i n - H e c k e

32 Z ( s , C 1) + Z ( s , C 2) - Z ( s , C 3) - Z ( s , C 4) - Z(s,C

5) - Z ( s , C

6) - Z ( s , C

7)

+ (ZZ(s, C8)+ZZ(s, C 9) - Z ( s , Ct0 ~K(S) ~(s) where

K = Q(~/2). Z ( s , C1) + Z ( s , Cz) - Z ( s , C3) - Z ( s , C 4) + ZZ(s, C5) - Z ( s , C6) + ZZ(s, C 7) - Z ( s , C 8) - Z ( s ,

C 9) - Z ( s , Clo)

~K(S) ~(s) where

K = Q(~//Z0). Z ( s , C 1) + Z ( s , CZ) - Z ( s , C 3) - Z ( s , C 4) - Z ( s , C5) + ZZ(s, C 6) - Z ( s , C 7) - Z(s,

C8) - Z(s,

=g" where

C 9 ) + ZZ(s, C10)

~K(s) ~(s)

K = Q(~//10). Z(s, CI) - Z(s, Cz) - 2Z(s, C3) + ZZ(s, C4) + Z(s, C5) - Z(s, C6) - Z(s, C7) + Z ( s , C8) - Z ( s , C9) + Z ( s , C10) ~K(S) ~(s)

where

K = Q(~//5). T h i s l a s t c a s e ( i n v o l v i n g the 6th r o o t of 5) is m o s t i n t e r e s t i n g

and a l l o w s one

to e x t e n d D e d e k i n d ' s m e t h o d f o r o b t a i n i n g f o r m u l a e f o r the c l a s s - n u m b e r

of a f i e l d

33

Q(~/a)

to higher degree fields.

We

should like to cite the paper by H a r v e y C o h n

("A n u m b e r i c a l study of Dedekind's cubic class n u m b e r

formula, " Journal of l~esearch

of National B u r e a u of Standards, Vol. 59, 4, 1957, 265-271) w h e r e the c l a s s - n u m b e r s of fields O(~/a)

are found for certain very large values of a.

W e would also like to record here that P r o f e s s o r Berndt has c o m m u n i c a t e d to us that he has verified the non-vanishing of the functions ~K(S) for fields

K = Q(~/a)

tioned above.

Serre

for a number only recentiy

at

1

s = Z

of values

of

a,

found an example

using the type of formula of a fieid

K

of degree

men8

for

which

~ K ( ~ ) = 0. In the above

list of "twists"

we omitted

4 cases,

T h e s e are left as an exercise

to the reader.

Dis c riminant

-18Z5Z

C 1 : (I, 0,4563)

CI0 = (19,8,241)

C Z = (27, 0,169)

Cll

C 3 = (4, Z, 1141)

CIZ = (37, I0, IZ4)

C 4 = (7, Z, 65Z)

C13 = (163,1Z, 73)

C 5 = (Z8, Z, 163)

C14 = (Z7, 18, 17Z)

C 6 = (Z8, Z6,169)

C15 = (43, 18,108)

C 7 = (49, 44,103)

C16 = (36, 30,133)

c8

C17 = (67, 30, 76)

= (9, 6 , 5 0 8 )

C 9 = (36, 6, IZ7)

= (31,

i0,148)

C18 = (61, 3Z, 79) C19 = (67, 46, 76)

Cn+17

= C

n

for

3 < n <

19.

G r o u p Table for Diseriminant

-18252

T h e r e are two generators, n a m e l y C 4 = (7,2,652)

and

C 9 = (36,6,127)

34

T h e group table is as follows: C4 z = C 7 3 C4

= C6

C44 = C 3

C 4 C 9 = C19

C 4 C 9 z = CIZ

Z C 4 C 9 = C14

C 4 2 C 9 z = C18

C44C9

C44C9 Z = C 8

5

C4

= C16

3

= C5

C46 = C z

3

Z

C 4 C 9 = Cll

C4 C 9

= C17

C 4 5 C 9 = CIO

C 4 5 C 9 Z = C13

T o complete the table, use

C n " ~ n =C I .

A character

f o r t h i s g r o u p c a n be d e f i n e d a s f o l l o w s : g~riatI

~ X (C4C 9) = e

lZ

Z~ri~tg

3

e

where t1 : 0,1,...,11

t Z = 0, 1, Z.

T h e r e are four twists into H e c k e L-functions of pure cubic fields, n a m e l y 7 Z ( s , C1) + Z ( s , CZ) + Z z

j=3

Z(s, C) J

~K(S)

19 Z ( s , cj) j=8 where

=

2

•

-

-

~(s)

K = Q(~/13).

Z ( s , C 1) + Z(S, Cz) - Z ( s , C3) - Z ( s , C4) - Z ( s , C5) + Z Z ( s , c 6) - Z ( s , c 7) - Z ( s , c 8) + Z Z ( s , c 9) - Z ( s ,

clo)

+ ZZ(S, Cll ) - Z ( s , C12 ) - Z ( s , C13) - Z(S, Cl4) + Z Z ( s , C15) - Z ( s , C16 ) + Z Z ( s , C17 ) - Z ( s , C18 ) - Z ( s , C19)

~K(S) ~(s)

35

where

K : O(~/Z).

Z(S, Cl) + Z(S, Cz) - Z(s, C3) - Z(s, C4) - Z(s, C5) + ZZ(s,06) - Z(s, C7) - Z(s,08) - Z(s, C9) - Z(s, CIo) - Z(S, Cll) - Z(s, CIz) + ZZ(s, CI3 ) - Z(s, CI4) - Z(s, 015) + ZZ(s, CI6) - Z(s,017 ) + ZZ(s, C18 ) + ZZ(s, CI9 )

tK(S) ~(s) where

§ 5.

K = Q(~/5Z).

In order to calculate the n u m b e r

of a given discriminant t{ber Zahlentheorie,

A w e need to quote, for example,

f r o m Landauts V o r l e s u n g e n

Bd. 1.

Satz Z13. Jedes und Z w a r

of primitive classes of binary quadratic f o r m s

ix - 0 oder

eindeuti~, = f m Z,

1 (rnod 4),

das Rein Q u a d r a t ist, is_t,

w___o m > 0 und

f Fundamental-diskriminante

ist. Satz ZI4.

E s sei A = f m Z die zerlegung unseres

&

nach Satz 213.

D a n n ist CO

K(a) : x (£)i: r=l

r r

n

{i- (l)!]K(f).

p[m

P P

Here ~W

K(f) -

h(f).

See Satz Z09, pp. 15Z and 141. h(f) denotes the n u m b e r binary quadratic f o r m s of discriminant Using these theorems, p

is a p r i m e

> 3,

of primitive classes of

f.

w e find that for the discriminant

the class n u m b e r

where

h(& ) is given by f

h(-108 pZ)

Ix = -i08 p Z

J3(p-l)

if p-I(3)

1

if p ~ Z(3).

3(p+l)

Looking back at the linear combinations of Epstein's zeta functions

Z(s, Cj),

36

that o c c u r r e d in Section 4, it is natural to ask for the n u m b e r Z(s, C.). This n u m b e r w h i c h is also the rank J zet{L functions of given discriminant

of linearly independent

p of the linear space of Epstein

Z A = -I08 p , is given by 1

p = [ + ~ h(-108 p2).

So

p : i0 for We

p : 5, p = 19 for

p =13.

This a g r e e s with our tables in §4.

note the following p h e n o m e n o n

the l i n e a r c o m b i n a t i o n s of the

which seems

w o r t h recording.

Z ( s , Cj) w h i c h a r e e q u a l to

2 •

Amongst

~K (s) ~(s-----~we o b t a i n

four combinations with

QI ' t,

o(b/pl,

Q (~.~'/Zp),

Q(~4p).

This has to do with the fact that the c o m p o s i t i o n g r o u p (of the primitive classes of binary quadratic f o r m s with discriminant index

3.

W h e t h e r the

K

w e w e r e unable to prove,

§ 6.

in

~K(s)

Z • ~(s)

-108 pZ)

has four distinct s u b g r o u p s of

is a l w a y s a sub-field of a K u m m e r

except in special cases above.

In o r d e r to p r o v e the a f o r e m e n t i o n e d assertions, w e develope s o m e

already m e n t i o n e d in a n earlier p a p e r (D. M . Kurnrner field f2 = Q(~/I, k~/a), w h e r e If p

field,

Goldfeld, loc. cit.). W e

a # + i or a perfect

is a rational p r i m e not dividing

ka

and

fl and

kth

ideas consider the

power.

fz are m i n i m a l

such

that

p then

p

fl _ = 1 ( n o d k),

x

k _ fz = a ( m o d p)

soluble,

is u n r a m i f i e d a n d factors in i2 as a product of

k~ (k) r

=

flfz flfz p r i m e ideals local factor

~51,~Z ..... ~r L

of P

i.e.

(p) = i51~Z...~r

~[3(s)' w e see that

and

N~i = p

. L o o k i n g at the

37

L

/ lI / 1 -

:

P pip ~

1

=

_

N@)s

flfzs P

Let

[l,~Z

be primitive

b

P

=

fl' fz th roots of unity, respectively.

fl

fz

]]

1]

hl=l hz=l

(

~

Then

Z

l - - -

Now define

~ 1 if x k ~a(p)soluble XP'k(a) :

e Z1ri/k if x k ~ a(p) .

Then -1 L

= P

because, as

X

l-

runs through the Dirichlet characters

k1 value ~l exactly value

i] 1] x ( m ° d k) w=l

~ (k 1) fl

times,

hz @Z exactly k / f 2 t i m e s .

~(s)

=n

Also Xp, k(a)

( m o d k),

(w=l,Z

. . . . .

X(P)

takes on

k) a s s u m e s each

Consequently

L (s)

P =

~v

P

-i

1] L(s, X) II 1] II X (rood k) w:l X (rood k) p

-

~' p

1

Let

e o = Q ($'/t)' K : Q(~,/a) . T h e n following Artin (fiber die Zeta Funktionen y o n algebraischer Zahlkt~rper, Collected Papers) w e get

1 ~ ~0)~, ~,

38

Cons equently

n

=

~(s)

n

i

i Li -

(6)

I w-I x(mod k) p

pS

k-

~7.

W e note that for discriminants

the positive integer n by

Cj

A = _3k Z,

the n u m b e r of representations of

(j fixed) is a linear combination of multiplicative

functions, which occur as coefficients in (6). This is in accordance with Hecke theory (he, however, uses the H e c k e operator, which does not appear in our work) according to which the n u m b e r of representations of n > 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(
The following question is suggested by the above work: combinations of Epstein zeta functions of a given discriminant s a m e functional equation) have Euler products? prime

> 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 <_a I < ... < a~ <_n, ~ = rk(n ) contains a n arithmetic p r o g r e s s i o n of k

terms.

44

Szemer6"di r e c e n t l y p r o v e d the old c o n j e c t u r e of T u r i n and m y s e l f (1)

rk(n) = 0(n).

(i) of c o u r s e contains V a n d e r W a e r d e n ' s t h e o r e m . for

r3(n ) are due to B e h r e n d

best

The

estimates

and IKoth w h o p r o v e d

n

c n Z r3(n) < loglog n

<

Cl(lOg n) I/Z e

T h e true o r d e r of m a g n i t u d e

of

rk(n)

is v e r y difficult to d e t e r m i n e .

I w o u l d ex-

pect that

rk(n)

=

0 (log n) ~'

holds for e v e r y

k

and

~ . A v e r y attractive conjecture of m i n e

states: Let 1 l < a I < a Z < ... be an arbitrary s e q u e n c e of integers with ~ - - = co . T h e n our a. i s e q u e n c e contains for e v e r y k an arithmetic p r o g r e s s i o n of k t e r m s . I offer 3 0 0 0 dollars for a proof o r d i s p r o o f of this conjecture. w o u l d i m p l y that for e v e r y There

k

there are

k

is an interesting finite f o r m

primes

is extended o v e r M 1

contain an arithmetic p r o g r e s s i o n of k G

< 0o,

in fact p e r h a p s a l r e a d y

u p p e r a n d l o w e r b o u n d s for

A k.

sequences terms.

A 3 = o0, I am

w h i c h do not

it w o u l d be v e r y desirable to h a v e g o o d

afraid u p p e r b o u n d s are h o p e l e s s at p r e s e n t I o b s e r v e d that A k >

k io~ Z 2

recently p r o v e d

(Z)

A k > (l+o-(l))k log k .

His proof will s o o n a p p e a r in Proc. may

a I < a Z < ...

It is not at all obvious that

so one should p e r h a p s concentrate on l o w e r bounds. and Gerver

Put

I a. l

i

the m a x i m u m

if true,

in arithmetic progression.

of our conjecture:

A k : max

where

T h e conjecture,

Amer.

Math.

Soc.

Gerver

believes that

be best possible. Perhaps

the following t w o further functions a r e of s o m e

interest:

Put

(Z)

45

1 a.

Ak,n,~ ~ = max a .
i

1

w h e r e the m a x i m u m

is to be taken over all s e q u e n c e s w h i c h do not contain a n arith-

m e t i c p r o g r e s s i o n of k

terms.

I of c o u r s e expect

interest to estimate f r o m a b o v e and b e l o w

. (n) = m a x /~k

where

the

for e v e r y fixed and

n

a.

Ak(n) < ck.

A k - Ak(n).

K a.>n i

It m i g h t be of

Define next

1 a. 1

do not contain an arithmetic p r o g r e s s i o n of k

terms.

i

I expect that A (n)

for k l i m a (n) = 0. It m i g h t be of interest to investigate A k - ~ k k n=o0 and large k. In particular is it 0(log n)? A l s o w h a t h a p p e n s if both k

n tend to infinity?

It is not clear w h i c h if any of these questions will lead to

fruitful results. A s e q u e n c e of integers

l < a I < ... < a k <_n

a. is fine arithmetic m e a n of other a' s. 1 T h e study of these s e q u e n c e s w a s We

is called n o n - a v e r a g i n g if no

started by E. Straus.

Put

maxk

= g(n).

have

(3)

e c(l°g n)I/Z < g(n) < n Z/3+E

T h e l o w e r b o u n d in (3) is due to Straus the u p p e r b o u n d to Straus a n d myself. 7

A b b o t t recently p r o v e d the u n e x p e c t e d to d e t e r m i n e Very

g(n) > cn I/I0.

It w o u l d be v e r y interesting

lira log g(n)/log n. n=o0 recently F u r s t e n b e r g p r o v e d S z e m e r ~ d i ' s t h e o r e m

by m e t h o d s

of

ergodic theory, his proof will be published soon. References I~. S z e m e r 6 d i , progression, of the p r o b l e m

O n sets of integers containing no

A c t a Arith. Z7(1975), Z99-345.

e l e m e n t s in arithmetic

F o r further literature a n d history

see I; II a n d the p a p e r of S z e m e r e d i .

]2. G. Straus, N o n a v e r a g i n g

sets, P r o c .

Syrup. P u r e Math.

P. E r d B s a n d E. G. Straus, N o n a v e r a g i n g Math.

k

AMS

sets II, Coll. Math.

1967. B61yai

Soc. C o m b i n a t o r i a l theory a n d its applications, N o r t h Holland A m s t e r d a m -

L o n d o n 1970 Vol. Z, 405-411.

46

Z.

Covering

A system

congruences

a n d related questions.

of c o n g r u e n c e s

(1)

al (rood n.l)'

is called a covering

system

l
if e v e r y integer satisfies at least one of the c o n g r u e n c e s

(i). T h e principal conjecture w h i c h is n o w m o r e can be arbitrarily large.

It is surprising h o w

500 dollars for a proof or disproof. system

with

n I = 20.

than 40 y e a r s old states that

The

difficult this conjecture is -- I offer

r e c o r d is still held by Choi w h o

= min >

___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
for w h i c h the density of integers not satisfying a n y of these c o n g r u e n c e s than or equal to

c,

Estimate

or d e t e r m i n e

the a s y m p t o t i c

properties

...

is less

of u (c) as n

c -- 0

and

n-- ~ .

Similar questions

I conjecture that for every n.1 a r e s q u a r e - f r e e

£

integers all w h o s e

can be a s k e d about

f(n, c) a n d

there is a covering s y s t e m prime

F(n,c).

(1) w h e r e

factors a r e greater than

p~ .

all the

47

Let

n I < n Z < ... < n k

ing to obtain conditions system

(I) exists.

0-(n)/n > C,

be a s e q u e n c e

of moduli.

(if possible n e c e s s a r y

In particular

but no s y s t e m

It w o u l d be v e r y interest-

a n d sufficient ones) that a covering

I conjecture that for every

(I) exists w h e r e

the

C

there is an

n

with

n. > 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<

if for e v e r y choice of residues

nz<

...

of rnoduli.

a . ( m o d n.) a n d to e v e r y 1

N

e > 0

h a s property there is a

k

1

so that the density of integers satisfying n o n e of the c o n g r u e n c e s

(3)

a.(mod

n.)

1

is less than

E .

N

1< i< k

1

is said to h a v e property

PZ

if there is a s e q u e n c e

of residues

a. so that the density of the integers satisfying n o n e of the c o n g r u e n c e s (3) is i k less than a . It has p r o p e r t y P 3 if this holds for a l m o s t all (i. e. 0( II ni) ) i=l choices of the residues a.. P 3 clearly holds if there is a s u b s e q u e n c e {nir } with 1 1 r ni r = 0o, (nirl, n irZ) = i, but at the m o m e n t I do not see a n e c e s s a r y a n d sufficient condition.

P2

certainly

d o u b t h o i d s if a n d o n l y if these

iines and must

trivial

1

1

is

equivalent

of integers O.

but it also holds if

I formulated

these problems

of the reader

to choose

with the condition:

which does not satisfy

On the other hand observe

so that every integer

suffices

1 iIE ~-i = ~ "

ask the indulgence

is clearly

the density

1< k < ~ a.

ia~.~-=l~i = ~ ,

if s o m e

n.1 = Zi"

Pl

no

whiie writing

of t h e q u e s t i o n s

are

or false. Pl

a.

h o i d s if

a. = i. 1

satisfies

For

every

choice of the residues

any of the congruences

that it is trivial

1

that one can find residues

at least one of the congruences

By a s l i g h t m o d i f i c a t i o n

ai(mod n.),

a.(mod 1

n.) - -

we can obtain a problem

1

which

48

is p e r h a p s

not trivial:

Let

n I < n Z < . . . w h a t is the n e c e s s a r y

condition that residues

a. exist so that all but a finite n u m b e r i satisfy one of the c o n g r u e n c e s

(4)

m

O n e can also ask:

--- a . ( m o d 1

W h a t is the n e c e s s a r y

integers satisfy one of the c o n g r u e n c e s

n.), 1

m

>

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<

Zd l

T h e choice

a set satisfying at least one of the c o n g r u e n c e s

a n d density

di, d 2 with

a. = 0 w o u l d then i with infinite c o m p l e m e n t

I. further questions c a n be a s k e d but I leave their formulation

to the

reader. A set of c o n g r u e n c e s integer satisfies at m o s t system

a i ( m o d ni) , n I < n 2 < ...

one of these c o n g r u e n c e s .

is called disjoint if every

I conjectured that no covering

can be exact i. e. every integer satisfies exactly one of the covering

congruences.

Mirsky

and Newman

v e r y s i m p l e proof of m y

Let

a i ( m o d hi),

be a disjoint system. as possible.

Put

Szemeredi

a n d R a d o found a

conjecture.

Stein a n d I asked:

(5)

a n d a little later D a v e n p o r t

i < n I < ... < n k < x

maxk

= g(x),

determine

or estimate

g(x)

as accurately

and Iproved

m1+ E

-Cl(lOg x) xe

Z

<

g(x)

<

x

c2 (log x)

We

believe that the l o w e r b o u n d is closer to the truth.

Szemer6di

and I tried

49

unsuccessfully to give n e c e s s a r y and sufficient conditions for a s e q u e n c e of m o d u l i n I < ... < n k

that a disjoint s y s t e m

A s far as I k n o w

a.(modl n.),1 1 < i < k

the following question w h i c h m a y

be of s o m e

interest has

not yet b e e n investigated:

L. for w h i c h (5) is a disjoint s y s t e m be

greater than

obably

m.

cm

L e t all th~ 1 = m a x Z n.

should exist.

1

em

--0

as

m--

oo. If true estimate

1

. Perhaps greater than

it w o u l d be better to require that all p r i m e factors of the

covering congruences.

sizes.

are

m.

T h e r e are m a n y

SchBnheim:

n

Let

~

recent generalisations of covering c o n g r u e n c e s and exact H e r e I only state a beautiful conjecture of H e r z o g a n d

kbe a finite A b e l i a n group.

HI,...,H. k

are cosets of different

P r o v e that i=~ H i n e v e r gives an exact covering of ~.

References

Summa

P. E r d ~ s , O n the integers of the f o r m Brasil Math. ii(1950), I13-IZ3.

P. E r d ~ s a n d I<. S z e m e r ~ d i , A r i t h m e t i c a 15(1968), 85-90. P. E r d ~ s , O n a p r o b l e m L a p o k 3(195Z), IZZ-IZ8.

Zk + p

On a problem

on s y s t e m s

and s o m e

related p r o b l e m s ,

of E r d ~ s and Stein, A c t a

of c o n g r u e n c e s

(in H u n g a r i a n ) Mat.

S. L. G. Choi, C o v e r i n g the set of integers by c o n g r u e n c e classes of distinct moduli, Math. C o m p . Z5(1971), 885-895.

Comp.

S. J. B e n k o s k i and P. E r d ~ s , Z8(1974), 617-623.

O n w e i r d a n d pseudoperfect n u m b e r s ,

Math.

50

3.

Some

applications

of covering c o n g r u e n c e s

In 1934 i%omanoff p r o v e d that the l o w e r density of integers of the f o r m 2k + p many

is positive.

He wrote me

w h e t h e r I can p r o v e that there are infinitely

o d d integers not of the f o r m

on covering c o n g r u e n c e s

2k + p.

and I proved

there is an arithmetic p r o g r e s s i o n w h i c h is of the f o r m

2k + p

This question led m e

to the p r o b l e m s

-- using covering c o n g r u e n c e s

-- that

consisting entirely of odd n u m b e r s

(this w a s p r o v e d independently

no t e r m

of

by V a n d e r C o r p u t

too). It is easy to see that if one could p r o v e that there a r e covering w i t h arbitrarily large metic progression most

r

n I then it w o u l d follow that for e v e r y

no t e r m

distinct p r i m e

of w h i c h is of the f o r m

T h e following question s e e m s

of

27

the f o r m

where

r

O

r

It is e x t r e m e l y

and

o d d integers not of

recently p r o v e d

is s q u a r e f r e e ?

~ > 0 there is a n r > r0(~ ) so that the l o w e r kI kr + ... + 2 is greater than I - ~ .

Let

PI' " " " " Pk

becomes

be the s e q u e n c e

f(n) the n u m b e r

of integers with

of solutions of f(n) > 0.

h o p e that the density of our s e q u e n c e of this type s e e m s

that

Zk + L

where

Is it true Z Pi ~ L

a n d let a I < a 2 < ...

In v i e w of i%omanoff's result one w o u l d exists.

to be far b e y o n d our resources.

n,f(n) > 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 <

q u e s t i o n c a n b e a s k e d for all

that if n_< a I < ... n

I conjectured

II): L e t

a r e distinct.

follows:

--0

determination

is a s e q u e n c e

p r o o f o r disproof.

for

possible.

exact

I and

Cbn

p a p e r in A c t a A r i t h m e t i e a ) .

I call attention to o n e of m y

in

an <

n

1 a r e all distinct t h e n

n i~=l~iai =

E

successful.

1 _< a.l < " " " < a n

or

where

n

satisfying

a n d Sullivan c a n solve p r o b l e m

I c o n j e c t u r e d a n d L e v i n e just p r o v e d then

same

in this p a 0 e r .

is only a little

a n d sufficient conditions for

b e s u c h that all the s u m s

n < log x + C ? log Z

I offer 3 0 0 dollars for a

this c o u l d h o l d w i t h

extremal

( w h i c h is of c o u r s e

problems

C = 3.

w h i c h I c a n not s o l v e state a s

...

b e a s e q u e n c e of i n t e g e r s for w h i c h all the s u m s a + a. -i j 1 a r e different. D e t e r m i n e max )2--. W e get different p r o b l e m s if i = j is ai p e r m i t e d o r not -- but I c a n not s o l v e a n y of t h e m . Let

a's.

a 0 = 0, a I = 1 < a Z < ... i Determine rain ~ - - . ai In s o m e c a s e s o n e e n c o u n t e r s

our sequence let sum

has density

a I < a Z < ... of t w o g r e a t e r

0

b e s u c h that e v e r y i n t e g e r is the s u m

problems

but it is m u c h

b e a n infinite s e q u e n c e a's.

of t w o

where

it is not h a r d to p r o v e that I ~-< 0% e.g. i ai of i n t e g e r s w h e r e n o a.i divides the harder

S~rkozi and I proved

to p r o v e that

that the d e n s i t y of s u c h a s e q u e n c e

53

is 0 but w e could not prove (I).

The

following

such

that

no

say,

if

finite

a. divides 1 x = 3n and the

1

~--< a. i

problem

o0

and are n o w h e r e near of settling p r o b l e m

remains

the

sum

of two

a's

are

the

here. greater

integers

Let a's.

1 <__ a 1 < . . . < a k <__ x Then

Z n , Z n + 1. . . .

k < [ 3 ] + 1. -,3n.

be

Equality,

References

P. Erdbs, P r o b l e m s and results in additive n u m b e r theory, Colloque sur la th~orie des n o m b r e s , Bruxelles. G e o r g e Thone, Li@ge; M a s s o n and Cie, Paris (1955), IZ7-137. P. E r d B s and A. S~rkozi, O n the divisibility properties of sequences of integers, Proc. L o n d o n Math. Soc., ZI(1970), 97-101.

54

5.

Some

more

extremal

problems

in additive a n d multiplicative n u m b e r

theory

S i d o n calls a s e q u e n c e of integers I <__a I < . . . a B k s e q u e n c e if the s u m s k ~.~ a ,s . = 0 or 1 a r e all distinct. S i d o n a s k e d in 1933: find B k s e q u e n c e for i=l i r i I which a tends to infinity as s l o w l y as possible. It is e a s y to see that t h e r e is n a B Z s e q u e n c e with a n < C n 3 for all n. O n e of the m o s t challenging p r o b l e m s here

states:

Is t h e r e a

B Z

sequence

with

a / n 3 -- 0 -- I give i00 dollars for a n

p r o o f o r disproof.

I of c o u r s e

e x p e c t that s u c h a s e q u e n c e exists -- in fact I a m Z+~ s u r e that t h e r e is a B Z s e q u e n c e with a < n for e v e r y E > 0 a n d n > n0(s ). n 2+~ P~gnyi a n d I p r o v e d b y probabilistic m e t h o d s that t h e r e is a s e q u e n c e a < n n

for w h i c h

f(m) =

~ 1 < c ai+aj=m ~

the o t h e r h a n d I p r o v e d

where

the c o n s t a n t

c

depends

that for e v e r y

B 2

sequence

n

= oo in fact

if { a n }

is a basis t h e n

An

n

l i m sup n=o0 n

n

(i) be i m p r o v e d ?

On

a

lim sup-~

Can

E .

E

a

(i)

only o n

> 0. log n

old c o n j e c t u r e of T u r i n

lira s u p f(n) = co,

more

and myself

generally:

let a

states that Z < c n D

n

n = l,Z,..,

is it then true that

lira s u p f(n) = 0o ?

I offered a n d offer 3 0 0 dollars

for a p r o o f or d i s p r o o f of these c o n j e c t u r e s . The interval. exceeding

B

sequences

Denote n.

behave

quite differently if w e

by

Bk(n)

the m a x i m u m

Turin

and I

(see also C h o w l a )

(l+o(1))n I/Z <

(l)

(i) is of c o u r s e

mentioned

in

I and

number

restrict t h e m

of t e r m s

of a

Bk

to a finite sequence

proved

Bg(n) < n I/2 + c n I/4 .

II.

We

conjecture

B z ( n ) = nl/Z + o(i) .

(Z)

I offer 3 0 0 dollars for a p r o o f o r d i s p r o o f of (2). Bose a b o v e for with Turan

and Chowla

k >__3. breaks

They

observed observe

down

and

that it is v e r y h a r d to e s t i m a t e B 3 ( n ) >__ (l+o(1))n I/3

B 3 ( n ) <__ (l+o(1))n I/3

Incidentally if a I < a Z < . . . that

lira s u p a n / n 3

= oo ,

but r e m a r k

Bk(n)

from

that o u r p r o o f

is open.

is a n infinite

B3

sequence

t h o u g h I h a v e no d o u b t that this is true.

I cannot prove

not

55

We

a r e v e r y far f r o m

as far as I k n o w

which

being able to solve p r o b l e m

there is not e v e n a reasonable

(I) for

Bk

sequences

--

conjecture.

L e v i n e in a recent letter to m e asked: Is there a B Z s e q u e n c e a n d a n 1 ~ i/2 ~ _ o0 o It follows f r o m m y results that for every 6 > 1 i ai (log ai)

s >

l

(3)

~ ai)i/Z(log )6 i (a i l o g l o g a. 1

But I h a v e an e x a m p l e

of a

B 2

< o0.

s e q u e n c e for w h i c h

1

(4)

z

I/z

i a. 1

=

~"

(log log a.) 1

In trying to close the gap b e t w e e n

(3) a n d

(4) L e v i n e asked:

Is it true

that

(ai log ai)i/Z(log log a.)1

converges

for every

I proved:

BZ

sequence?

There

exists a n infinite

BZ

s e q u e n c e with

Bz(n)

(5)

lira s u p

n=0o

i/2

in

(5) w a s

improved

possible result could be

to I/Z I/Z i.

by K r u c k e b e r g .

a I < a Z < ... < a k

modulus

a n d a perfect difference set There

is an integer

b l , . . . , b u + 1 so that every residue the

a's

occur amongst

the

be a n y

m

= u

mod

BZ

rood m Z

sequence.

m

T h e n there exists a

w h i c h contains the

+ u + 1 and

seem

like this h a v e b e e n investigated a n d in s o m e

structures.

(I) the best

u + 1 residues

is uniquely of the f o r m

a' s.

In

rood m ,

b.1 - b.j a n d

b's.

This conjecture if true w o u l d

a n d others,

In v i e w of

This w o u l d follow if the following conjecture of m i n e

Let

m

,

n

w o u l d hold:

other w o r d s :

1

--Ui-/z _>~

for Steiner s y s t e m s

to m e

to be v e r y interesting.

cases solved by T r e a s h ,

a n d other m o r e

Questions

Lindner

complicated combinatorial

0

56

References P. ErdBs and A. P~nyi, Additive properties of r a n d o m sequences of positive integers, Acta Arith 6(1960), 83-110, see also Halberstam-P~oth, Sequences, Oxford Univ. Press, 1966. P. ErdSs and P. TurAn, O n a p r o b l e m of Sidon in additive n u m b e r theory and on s o m e related problems, Journal L o n d o n Math. Soc. 16(1941), Z12-Z16, A d d e n d u m 19(1944), 208. A. Stbhr, Gelbste und ungelbste F r a g e n ~iber B a s e n der nat~irlichen Zahlenreihe I, II J. reine a n g e w Math. 194(1955), 40-65, ili-140. This paper has m a n y p r o b l e m s and results and a very extensive bibliography. It contains the proof of (i) and (5). C. Treash, T h e completion of finite incomplete Steiner triple systems with applications to loop theory, J. Combinatorial Theory, Set A 10(1971), Z59Z65, for a sharper result C. C. Lindner, E m b e d d i n g partial Steiner triple systems, ibid 18(1975), 349-351.

Math.

F. Kr[{ekeberg, B z - F o l g e n und verwandte Zahlenfolgen, J. reine a n g e w Z06(1961), 53-60.

57

6. Graham

Problems

on infinite subsets

a n d Rothschild conjectured that if w e

classes then there a l w a y s is a n infinite s e q u e n c e

split the integers into t w o

aI < a 2 < ...

so that all the

finite s u m s

(i)

52ekak ,

a r e in the s a m e

ek = 0

or

1 .

class.

This conjecture w a s fied by B a u m g a r t n e r .

proved

recently by H i n d m a n

a n d the proof w a s

simpli-

I just h e a r d that G l a s e r using a n idea of Galvin obtained a

v e r y interesting topological proof of the t h e o r e m . A f e w days a g o I asked:

Is there a function

f(n) so that if w e

integers into t w o classes there a l w a y s is a s e q u e n c e holds for infinitely m a n y

n

Galvin just s h o w e d splitting as follows:

Let

a n d so that

that no s u c h F(m)

in the first class if y > F ( x )

-- o0

a I < ...

split the

for w h i c h

an < f(n)

(I) holds ? f(n) exists.

T o see this he defines a x sufficiently fast. P u t n = g y, y odd. n

a n d is in the s e c o n d class if y < F(x).

is

It is easy

to see that this construction gives a counter e x a m p l e . There problem. k

(or

m i g h t be t w o w a y s

to save the situation a n d obtain s o m e

Is it true that there is a n

N0)

classes there is a s e q u e n c e

one of the classes is disjoint f r o m ask a weaker classes,

statement:

i.e.

{A

}

The

ekXk

{x }

of p o w e r

c

a n d the s u m s

x + y = z

of S ?

S, y ~

S

sequence

of S

a l m o s t disjoint and

co c {an},

all the classes

is

A

into t w o classes.

so that all the s u m s

Let

Sx

be a set of real Is there then a set

so that all the s u m s

If the a n s w e r

is no then w e

{Xc~1 + x a 2 }

also

could p e r h a p s a s s u m e

a r e distinct.

V a n d e r !~aerden's t h e o r e m .

of positive density.

n

is not solvable in S.

I thought of strengthening H i n d m a n ' strengthened

(i) do not m e e t

Split the real n u m b e r s

Is the following true:

in the c o m p l e m e n t

belong to the c o m p l e m e n t x + y, x (

many

1 < ~ < co (A0zlf-hA~z) < ~ 0 -c then there is an infinite s e q u e n c e

f(n) for infinitely m a n y

class?

so that the equation

that all the

n

{Xn)X n<

a r e in the s a m e

numbers

One would even

is a set of integers

s e c o n d possibility w o u l d be:

Is there a s e q u e n c e

~kak?

Divide the integers into c o n t i n u u m

< f(n), for infinitely m a n y

nontrivial

split the integers into

a I < a Z < ..., a n < f(n) so that at least

the set of all s u m s

the initial ordinal of the continuum, a

f(n) so that if w e

s theorem

in the s a m e

Is the following true:

Is there an infinite s e q u e n c e

w a y as S z e m e r 6 d i Let

A

a I < a Z < ...

be a a n d an

?

58

integer

t

observed

so that all the integers

a. + a. + t a r e in the s a m e c l a s s ? Straus l j that the full s t r e n g t h of H i n d m a n ' s t h e o r e m d o e s not h o l d in this case.

Some

t i m e a g o I thought of the following fascinating possibility:

the i n t e g e r s into t w o classes.

Is it true that t h e r e a l w a y s

Divide

is a s e q u e n c e

al, a 2 ....

so that all the finite s u m s same

~ ¢ .a. a n d all the finite p r o d u c t s II a. a r e in the x I i I A t this m o m e n t the p r o b l e m is open. M o r e g e n e r a l l y o n e c a n ask:

class.

Is t h e r e a n infinite s e q u e n c e formed

from

answer

is n o but no c o u n t e r e x a m p l e The

the

a's

a I < a Z < ...

a r e in the s a m e

following m u c h

a I < a 2 < ...

weaker

so that all the m u l t i l i n e a r

class?

One

would

perhaps

expressions

guess

that the

is in sight. c o n j e c t u r e is also open:

Is t h e r e a s e q u e n c e

so that all the s u m s

class?

Perhaps

Graham

proved

we

a. + a and products a a. a r e in the s a m e i j 1 j s h o u l d also r e q u i r e that the a a r e also in the s a m e class. 1

that if w e

four distinct n u m b e r s Hindman

proved

divide the integers

x, y, x+y, x y

that if w e

divide the integers

four distinct n u m b e r s

the s a m e

S o far nothing is k n o w n

Answering

there always

(i=j p e r m i t t e d )

a r e in the s a m e

Hindman

just i n f o r m e d

that this will b e c o r r e c t e d found a decomposition sequence

exists.

h a s density

theory),

is a n infinite s e q u e n c e

0.

class - Z 5 Z is best possible.

Z < t < 990

all g r e a t e r than in c a s e w e

a q u e s t i o n of E w i n g s ,

the J o u r n a l of c o m b i n a t o r i a l

into t w o c l a s s e s t h e r e a r e

all in the s a m e

there are always class.

< Z5Z

Hindman

that if w e

into t w o classes, 1

x, y, x+y,

assume

proved

xy

then

all of

all the i n t e g e r s

>__3.

(will a p p e a r s o o n in

divide the integers into t w o c l a s s e s

x I < x Z < ...

so that all the s u m s

x

i

+ x. J

class.

me

that t h e r e m a y

b y the t i m e this p a p e r

into t h r e e c l a s s e s

In fact H i n d r n a n

b e a g a p in his proof, appears.

A[, A Z, A 3

observes

but I h o p e

O n the o t h e r h a n d h e

so that no s u c h infinite

that o n e of his s e q u e n c e s

say

A 1

In his e x a m p l e

(i)

Al(X ) =

~

1 <

c x I/2 .

ai~ A 1 a,<x 1--

It is not yet clear w h e t h e r More

than I0 y e a r s

(and to w h a t a g o P~. L.

extent) Grahaln

(1) c a n b e i m p r o v e d . and I conjectured

that if w e

the i n t e g e r s into t w o c l a s s e s t h e n

(2)

1 = 1Z !x i

'

Xl < x 2 < "'"

(finite s u m )

split

59

is a l w a y s solvable with the

x. all in the s a m e class. This should probably not 1 generalisations are possible.

be too difficult. Clearly m a n y

References

n,

Neil H i n d m a n , Finite s u m s with s e q u e n c e s within cells of a partition of J. C o m b i n a t o r i a l T h e o r y Ser A 17(1974), i-ii, J. B a u m g a r t n e r , ibid. 384-386.

80

7. Let

1 _< a I < ... < an

minimum

number

r

a' s.

of the

A new

extremal problem

be a s e q u e n c e

of integers.

of distinct integers w h i c h are the s u m

I conjectured that for every

r and

If true this s e e m s

difficult a n d s e e m s

extremely

E > 0

D e n o t e by

fr(n)

the

or product of exactly

if n > n0(E , r), fr(n) > n

r-E

to require n e w ideas

(unless of c o u r s e an obvious point is being overlooked). Szemer6di

and I observed

that it follows f r o m

d e e p results of F r e i m a n

that

~=~fz(n)/n

but even the proof of fz(n) > n

i+¢

= ~,

seems

to p r e s e n t great difficulties.

Z fz(n) >

n -

-

(log

is certainly false.

Perhaps

n

D e n o t e by

F(n)

2

n) k

the true o r d e r of m a g n i t u d e

of fz(n)

is

exp (-c log n/log log n) .

the smallest integer so that there are at least

integers w h i c h are the s u m

or p r o d u c t of distinct

a.'s.

It s e e m s

F(n)

distinct

certain that

I

F(n) > 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 <

problems

...

on primes

of integers

satisfying

A(x) =

~ 1< a.<x 1

log x

so that all sufficiently l a r g e i n t e g e r s a r e of the f o r m p + a. ?

If this i s

1

i m p o s s i b l e then p e r h a p s such a sequence exists for which the density of i n t e g e r s not of the f o r m p + a. is O. Clearly many s i m i l a r questions can be asked for i o t h e r s e q u e n c e s then the p r i m e s but t h e r e a r e v e r y f e w results. Ruzsa proved that t h e r e is a s e q u e n c e integer is of the f o r m

of i n t e g e r s Zk + a .

a I < a Z < .... A(x) < c x / l o g x

so that e v e r y

Is it true that t h e r e is s u c h a s e q u e n c e

for e v e r y

1

c<

log Z +

E

The

prime

integers w h i c h

k-i~ple conjecture

d o not f o r m

t h e r e are infinitely m a n y primes.

This problem

infinite s e q u e n c e

states:

a complete integers

Let

aI <

set of r e s i d u e s

n

rood p

so that all the integers

is u n a t t a e k a b l e at present.

of integers.

... < a~

It w o u l d

Let

are prime.

Perhaps

it w o u l d

for a n y

p.

n

reasonable

are

be an

b e interesting to find a n e c e s s a r y

be more

Then

{n+ai} I < i < k

a I < a Z < ...

sufficient condition for the e x i s t e n c e of infinitely m a n y n + a.

be a set of

and

so that all the integers to p e r m i t

for e a c h

n

a

1

finite n u m b e r

of exceptions.

should be mentioned

here:

An Are

so that all but a finite n u m b e r number

of p r i m e s

unfortunately

old a n d v e r y fascinating c o n j e c t u r e there two sequences

of the s u m s

a

a r e of the f o r m

nobody

can prove

aI < a Z <

of O s t m a n

... ; b I < b Z < ...

+ b . a r e p r i m e s a n d all but a finite z 3 T h e a n s w e r is o b v i o u s l y No'. but

a. + b ? i 3 this. H o r n f e c k

showed

that both s e q u e n c e s

must

b e infinite. Let us now

r e t u r n to o u r p r o b l e m .

If n

is s u c h that all the integers

n + a., i = 1,2 .... a r e p r i m e s w e first of all clearly m u s t h a v e (n+a.) ~ (n+a.) i 1 j a n d w h a t is m o r e (n+ai, n+aj) = i. Is it p o s s i b l e to find a n e c e s s a r y a n d sufficient condition for the following t h r e e p r o p e r t i e s

of a n infinite s e q u e n c e

is a n o t h e r infinite s e q u e n c e

(ai+bj) ~ (ar+bs) ,

B

so that

i.

3.

a. + b. a r e all p r i m e s ? I think p r o p e r t y i 3 but I h a d n o t i m e to think this o v e r carefully. reasonable

conjecture.

The

problems

may

1 and For

change

There

(a.+b.,a + b ) = i, I 3 r s can probably be handled,

Z 3

A?

Z.

the only h o p e w o u l d

if w e

permit

for e a c h

be a a. 1

b. a finite n u m b e r of e x c e p t i o n s also w e c o u l d restrict o u r s e l v e s 9 assume (a.+b. , a.+b. ) = i for i <_jl < Jz < o0.

1

De

(l)

J1

Bruijn,

z

JZ

Turin

and I considered

f(n)

=

z

,.i

the function

. . . . .

p < n n-p

to asking:

and

63

It is not difficult to s h o w

(z)

that

!

x

It follows

from

z

f(n)-l,

± ~ x

n< x

fZ(n)-l.

n< x

n l r ( n + n ~) - lr(n) > c 1--oog n

Hoheisel'sclassical

that

l i m inf f(n) > 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) <

One

lira sup f(n) = o0 n=o0

in fact feels that

~ ~(x-y).

~r(x) - it(y) should be usually of the o r d e r

of m a g n i t u d e

x-y a n d therefore it is r e a s o n a b l e to g u e s s that (4) is satisfied for e v e r y log x y < x - (log x) C for sufficiently large C. In fact I c a n not at this m o m e n t disprove:

x-y w(x) - ~(y) < c I log x

(5)

(5) w o u l d

imply

f(n) < c logloglog

n

could try to study

f(p)

y < x-

(log) C

and perhaps

li---~m ( n ) / l o g l o g l o g

We

for

n > 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 <--m

be the s e q u e n c e

of

(n,m) k

1

<

i=l qi -n

for a certain absolute constant

c.

~ p<m-n

1 --+c p

Trivially the opposite inequality is not true,

since there a r e arbitrarily large gaps b e t w e e n the p r i m e s .

It does not s e e m

to

be trivial to p r o v e that

(8)

lira inf ( ~ m

n=o0

z

n< qi < m

I

- -

qi-n

-

A)

x

p< m

= _~

p

A t p r e s e n t I do not see h o w to p r o v e (8). Eggleton, problems

Selfridge a n d I a r e writing a long p a p e r on s o m e w h a t

in n u m b e r

of our p r o b l e m s

theory.

(9)

We

conjecture

g(n) -- oo as

a k < k Z÷s

for

Let

(n-ak, n-at) = 1 for all

g(n) =

only p r o v e

(i0)

O u r p a p e r will a p p e a r in Utilitas M a t e m a t i c a .

related to (I) states as follows:

smallest integer for w h i c h

unconventional

n - - oo.

a 0 = 0, a I = I, a k

0 < i < k.

is the

Put

l ~ a. i=l I

This is p r o b a b l y v e r y difficult.

k > (log n) C, C = C(~),

a k < C k log k if k > (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
(13)

Is it true that for infinitely m a n y

k Z a k > 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 <__a I < . . .

(0 < c Z < Cl)

w(n) + c Z n 3 //4(/l o g

(i)

Let

To

Assume

c

n3/~

Let

/n 3/4/ ) og n) 3/Z + o \ / (log n) 3/Z

I <__a I < . .. < ak(n) <__n

be a s e q u e n c e of

that

lauav - atasl >l for every choice of the indices prove

u,v,t,p.

Does

(i) r e m a i n true?

I cannot even

k (n) = o(n). Clearly nearly all the e x t r e m a l p r o b l e m s

during m y

long life can be extended in this way.

munbers

or m o r e

lhil < n

be

n

generally m e m b e r s

complex

numbers

in n u m b e r

In fact the a's

of a vector s p a c e

assume

theory w h i c h I c o n s i d e r e d

e.g.

could be c o m p l e x

Let

h I ... h k,

that

lhahb - hchdl ~ 1

holds for e v e r y

1 5 _ a , b , c , d <_k.

but at this m o m e n t

How

large can be

If the h's

I do not see it.

max

k ?

o(n Z) is really certain

are c o m p l e x integers a result like

(Z) can undoubtedly be proved. Now

I discuss s o m e

more

such questions.

A s e q u e n c e of integers

a I < a Z < ...

is called a primitive s e q u e n c e if a i ~ a.. Primitive s e q u e n c e s h a v e 3 b e e n investigated a great deal see e.g° our s u r v e y p a p e r with S~rk~zi a n d S z e m e r e d l . I

But p r o b l e m problem an

e > 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)<_~(x)

?

a .<x 1--

(I) is certainly a fascinating conjecture. Beurling p r i m e

numbers

The

a's

are sometimes

a n d h a v e a large literature - as far as I k n o w

n e v e r b e e n c o n s i d e r e d before.

A v e r y nice a n d unpublished

called (i) has

conjecture of Beurling

69

states: A s s u m e

that the n u m b e r

is x + o(log x).

T h e n the a's

l h a v e to apologize if m y not always r e m e m b e r

of n u m b e r s

satisfying

not exceeding

x

references are s o m e t i m e s inaccurate but one does things

O n c e the following beautiful p r o b l e m w a s

D o e s there exist an infinite sequence of distinct G a u s s i a n p r i m e s I did not r e m e m b e r

T h e conjecture w a s told m e

theory m e e t i n g 1963 N o v e m b e r Motzkin.

i

w h o suggested a p r o b l e m in a discussion w h e r e m a n y

[Yn+l " Y n I < C ?

cleared this up.

IIa. i

are the primes.

are m e n t i o n e d in rapid succession. attributed to m e :

of the f o r m

w h o told m e

this but E. Straus

by M o t z k i n at the P a s a d e n a n u m b e r

and it w a s apparently raised by Basil G o r d o n and

I naturally liked it very m u c h

and told it right a w a y to m a n y people,

naturally attributing it to Motzkin, but this w a s later forgotten.

T h u s the p r o b l e m

is returned to its rightful owners. T h e following p r o b l e m w a s considered by G r a h a m S

be a m e a s u r a b l e set in the circle

two points of S

is an integer.

How

S~rkBzi and myself:

Let

IYl < r and a s s u m e that no distance b e t w e e n large can be the m e a s u r e

of S ?

S~rkSzi

has the sharpest results, but nothing has been published yet.

References P. ErdSs, O n s o m e applications of graph theory to n u m b e r theoretic problems, Publ. l%amanujan Inst. i(1969), 131-136, see also S o m e applications of graph theory to n u m b e r theory, T h e m a n y facets of graph theory Proc. Conf. W e s t e r n M i c h i g a n Univ. K a l a m a z o o 1968 Springer Verlag, Berlin 1969, 77-8Z. P. Erd}Js, A. S~rkSzi and E. Szemer4di, O n divisibility properties of sequences of integers N u m b e r theory Colloquium Bdlyai Math. Soc. N o r t h Holland Z(1968), 36-49.

70

I0. Let the number

Some

more

unconventional

problems

a. < a. < ... b e a n infinite s e q u e n c e 1 I of solutions of

of integers.

Denote

by

f(n)

V n

=

~

a.

Is there

a sequence

rightly

for which

criticized

seems

very

as

being

strange

and

f(n) -- 0o as

artificial

attractive

and

. 1

i=u

n--

0o.

This

problem

in the backwater

to me.

If

a. = i

can

perhaps

of Mathematics

then

f(n)

be

but it

is the number

of odd

1

divisors

of

n.

probabilistic

Iknow

of no

methods

Leo

example

but they

Moser

and

where

do not

I considered

f(n) >__ Z

seem the

for

all

n > nO.

I tried

to work.

case

where

the

a.

are

primes.

We

con-

1

j e c t u r e d that We

[ i m f(n) = 0o

d o not e v e n k n o w MacMahon

(1975), 9 2 2 - 9 2 3 ) sequence

a n d that the density

that the u p p e r

and Andrews consider

x.'s

(i. e.

density of the integers w i t h

(see G.

E. A n d r e w s ,

the following p r o b l e m .

of integers w h e r e

consecutive

xn U

x

Iknowit

I could Let

xI < x Z <

is not

Andrews

= (1+o(1))

even

not settle ...

be

known

f(n) > 0

Math

(lower

that no

density)

is positive.

Monthly

32

b e the

is not the s u m

of

that

n log n loglog n

x

/

- all I could x

is the

Jn

sum

-- o0 . do is to ask

a few

of consecutive

other

x.'s.

n

that the density

exists.

1 = x I < x 2 < ...

conjectures

whether

this question

such

Let

f(n) = k

1

n

far as

Amer.

is the s m a l l e s t integer w h i c h

x n ~ ~x.).

1

As

of integers w i t h

questions.

Is it true

1

of the

x.'s

is

O?

I am

not

sure

about

the density

1

but would

be very

surprised

if the

lower

density

would

not be

O.

v

Assume the density that

x

now

of this

that all the sequence

> c n log n must

is

hold

sums O.

Ex are u i It is obvious

for infinitely

distinct.

I am

by a simple

many

n,

thus

now

averaging

the lower

confident

that

process density

is

O.

n

It is not hard

to show

that for these

sequences 1

E Z n<x

--< x .1

C


i ]

holds

for

to prove

an absolute or disprove

constant this.

C.

Perhaps

Z --=-~ X. 1

converges

but I do not see how

71

Let

x I = 1 < x Z < ...

v ~x.

sums

u

I am

Xn

is the s m a l l e s t integer for w h i c h

a r e distinct.

A

simple

counting a r g u m e n t

gives

x

finite a n d infinite I am

have time

cn

for all

n.

B Z

3 -- 0. All the p r o b l e m s on n c a n b e a s k e d h e r e too, but a l m o s t nothing is

sequences

not really s u r e h o w

more

question:

I < ... < a k ~ n

consecutive

a's.

Hofstadter

of i n t e g e r s

then

consecutive

sequence,

m,

n<

m~

L(n) <

L(n)

integer so that for e v e r y

which

is not the s u m

s e v e r a l of his p r o b l e m s sample

as follows:

What

a I = 1, a Z = Z.

is the a s y m p t o t i c

of his p r o b l e m s :

Let

of the s e q u e n c e ,

behavior

a I = Z, a 2 = 3.

subtract

Does

were

inspired

Define a

If a I < ... < a n

is the s u m

of t w o

of this s e q u e n c e ?

Form

1 and append

l~epeat this o p e r a t i o n indefinitely.

which

of his p r o b l e m s :

a n + l is the s m a l l e s t i n t e g e r w h i c h

a's.

of

C n?

is a s m a l l

a I < a Z < ...

a r e a l r e a d y defined,

distinct e l e m e n t s

b e the s m a l l e s t

told m e

Here

Xn/

carefully.

L(n)

Is it true that

b y this q u e s t i o n of U l a m .

Another

Let

t h e r e is a n

l~ecently D.

with

difficult t h e s e q u e s t i o n s a r e since I did not

so far to investigate t h e m

One

or more

< n

s u r e that t h e r e is s u c h a s e q u e n c e

sequence

all the

3 1

known.

l~a

where

all p r o d u c t s

of t w o

these elements

to the

this s e q u e n c e

h a v e positive

density? A f e w d a y s ago, a sequence

1 ~ a I < a 2 < ...

a 2 ..... a k number

Kenneth

are already

a

a I = i, a n d the n u m b e r

Let

+ a. < x. 1

x

x).

Then

is a l w a y s

put

> x.

1 < i < j < k

of solutions of

hoped

a

+ a

which

would

Unfortunately,

a. + a. < x is less t h a n 1 j-that this a n d a g o o d deal m o r e is true.

Let is a n l
al,...,a n nO
that the

(or a 0 = 0,

< x,

0 < i < j < k, j > 0 of solutions of

of solutions of

show we

that m y

There

considered

theorem

J

--

with Fuchs

is

that the n u m b e r

is little doubt,

though,

the following question:

Is it true that t h e r e v s o that for n > n O the n u m b e r of distinct s u m s of the f o r m .~ l=u ai' n, is less t h a n ~ n Z ? W e p r o v e d this if a. =i, but c o u l d not attack i

the g e n e r a l

case.

of the i n t e g e r s

a. + a. < x i j-< x will then

a. + a

c o u l d not e v e n p r o v e

x + o(x).

(of 1976) H a r h e i m

be a permutation

is less

i--

that the n u m b e r

that the n u m b e r

of solutions of

in S e p t e m b e r

x - k

1

x + o ( x I/4+~)

essentially best possible.

Early

is less t h a n

/

b e of the f o r m

assume

Define

j--

x = a k + I. O b s e r v e

Rosen

--

a I = i,

b e the s m a l l e s t i n t e g e r for w h i c h

1

than

the following construction:

inductively as follows:

defined.

of solutions of

l~osen told m e

l,Z,...,n.

72

References P. ErdBs and W. H. J. Fuchs, O n a p r o b l e m of additive n u m b e r J. L o n d o n Math. So¢. 31(1956), 67-73.

theory,

EXPLICIT

FORMULAS

IN THE T H E O R Y OF A U T O M O R P H I C

FORMS

C. J. Moreno.

CONTENTS

Introduction

§ i.

§ 2.

§ 3.

. . . . . . . . . . . . . . . . . . . . . . .

Automorphic

Representations

and Euler P r o d u c t s

. . . . . . . . . .

i.i.

Classical

1.2.

Automorphic

1.3.

Relation

1.4.

Langlands'

1.5.

The F u n c t i o n a l

1.6.

Some E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . .

Rankin's

Automorphic Forms

Between

75

81

Forms . . . . . . . . . . . . . . . . .

on Adele Modular

81

Groups . . . . . . . . . . . . . .

Forms

and Forms on Adele

Groups

8Z . .

Euler P r o d u c t s . . . . . . . . . . . . . . . . . . Equation

Convolution

2.2.

The Constant

Term Matrix

Group

. . . . . . . . . . . . . . . . . . . . . . . .

2.3.

Some Euler P r o d u c t s

2.4.

The A v e r a g e

Zeros

in the Critical

3.1.

The H a d a m a r d

3.2.

The First v o n M a n g o l d t

3.3.

Explicit

and

for the E i s e n s t e i n

their F u n c t i o n a l

Size of the E i g e n v a l u e s

Series

Equations

of H e c k e

99 99

of the 109

.....

Operators

. . .

Strip . . . . . . . . . . . . . . . . . . . .

Product

Estimates

Formula

. . . . . . . . . . . . . . . .

Formula

9Z 95

Method . . . . . . . . . . . . . . . . . . . .

The I n g r e d i e n t s . . . . . . . . . . . . . . . . . . . . . . .

Fo(N)

86

of Euler P r o d u c t s . . . . . . . . . .

2.1.

85

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

119 131

148 148 15Z 158

74

§ 4.

§ 5.

§ 6.

Explicit

Formulas . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.

The Second

4.2.

Examples

von M a n g o l t

. . . . . . . . . . . . . . .

[70

Formulas . . . . . . . . . . . . . . . .

181

L(s,~) . . . . . . . . . . . . . . . . . . .

184

of Explicit

Zero Free Regions

for

5.1.

A Hadamard-Landau

5.2.

Prime N u m b e r

5.3.

The P r o b l e m

Type

Formula

170

Inequality . . . . . . . . . . . . . .

Theorems . . . . . . . . . . . . . . . . . . . . of E x c e p t i o n a l

Zeros

. . . . . . . . . . . . . .

184 193 197

Zeta D i s t r i b u t i o n s

. . . . . . . . . . . . . . . . . . . . . . . .

199

6.1.

Introduction

. . . . . . . . . . . . . . . . . . . . . . . .

199

6.2.

The E x p l i c i t

Formula

Z0[

References

and W e y l

Symmetry

. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZI5

75

Introduction

The purpose of these notes is to study the analytic properties products associated

to automorphic

adele ring of the rationals. classic trilogy

[ 30 ].

representations

where

~

function

T(n)

does not vanish on the

Secondly it is shown that the Euler product asso-

ciated with

T(n) 2,

continuation

to the whole s-plane with a pole at

at first only defined for

Re(s) > ii, s = ii.

about the average behavior of

has a meromorphic Lastly it is shown

T(n) 2

gives some informa-

T(n).

In the years following the publication

of Rankin's work, only his third

paper received much attention becaus~ of its connection with Ramanujan's ture

IT(p) l < 2p II/2.

at least for automorphic

representations

have received little attention up till now.

GL2~A),

The first paper seems to

It is our intention in these notes

to extend these results of Rankin to the Euler products tations and to develop applications

that the full sig-

of the adele group

of Rankin's second paper seems to have been realized.

of automorphic

represen-

to a circle of ideas that have their origin

in the proof of the prime number theorem. applications

conjec-

It is only in recent years, when harmonic analysis has

yielded a novel point of view to the theory of modular forms, nificance,

is the

There it is first shown that the Euler product associated

line of absolute convergence.

tion about the size of

GL2~),

The genesis of this work is to be found in Rankin's

to the well known Ramanujan arithmetical

that good information

of

of Euler

The results we obtain have important

to the study of the arithmetical

properties

of automorphic

represen-

tations.

To set the stage for our subsequent discussion, main results for the classical

we now explain Rankin's

theory of modular forms.

The Euler product associated with the function

~(s) = (2~)-SF(s)

~ P

T(n)

is

(i - T(p)p -s + pll-2s)-l,

76

where

T(n)

is the

n-th coefficient of the q-expansion oo

q ~

co

(i - qn)24 =

n=l

In

[ 30 ]-I

[ T(n)qn. n=l

Rankin proved, by a meLhod that is also applicable to the Euler

products of all holomorphic cusp forms that are eigenfunctions of the Hecke operators,

that

~(s)

does not vanish on the line

Re(s) = --.~

In these notes

we show that indeed Rankin's method applies to the more general class of Euler products associated to cuspidal automorphic representations

of

GL2~A); further-

more, we will obtain zero-free regions which are similar to those that arise in the study of Riemann's zeta function already exploited by Rankin in

~(s).

[ 30 ]-II,

A key idea in our >roof is the fact, that the coefficients of the Dirichlet

series associated to the Euler products satisfy the Petersson-Ramanujan a fact now established by Deligne resentations of

GL2~A)

conjecture,

[ 7 ] only for those cuspidal automorphic rep-

over the rationals whose component at infinity belongs

to the discrete series, on the average.

In particular we prove a result, well

known to experts but not stated explicitly in the literature,

that implies that

the coefficients in the Fourier expansion of a real analytic cusp form satisfy the Petersson-Ramanujan

conjecture on the average.

The proof of this result is

along the same ideas as in [ 30 ]-III, and the crucial point is the possibility of obtaining sufficient information about the analytic nature of the Euler product associated with the Dirichlet series n~la(n)2n -s. had done in [ 30 ]-II

when the

a(n)

This is indeed what Rankin

are the Fourier coefficients of a holo-

morphic cusp form which is an eigenfunction of the Hecke operators; respect it must be proved that the Euler product for real point on the line of absolute convergence, from the point of view of hamonic analysis, of the real analytic Eisenstein series at

a(n)

2

in this

has a pole at the

and this is seen, in retrospect

to depend on the distinguished pole s = 1

and the fact that the Petersson

inner product of a cusp form with itself is essentially the residue of the Dirichlet series of

a(n) 2

at the pole.

cuspidal automorphic representations

~

More precisely we prove that given two and

~'

of

GL2~A) one can associate an

77

Euler product

L(s,g×7')

whose local factors are 3-dimensional,

be the Euler product of an automorphic a meromorphic

continuation

representation

GL3(/A), [ ii ],

and has

to the whole s-plane and has no zeros and at most one

simple pole on the line of absolute convergence. representation

of

that seems to

associated with

When

T(n), L(s,~ ×~)

~ = ~'

is the automorphic

is, after a trivial normalization,

oo 2 -s n~iT(n) n .

the Euler product of

Other results that we prove are generalizations mulas of number theory.

of the von Mangoldt

In particular we study the distribution

Euler products on the critical strip and also prove various formulas like the following

for the Ramanjuan

function

1"'#(2)(0)

log

for-

of zeros of the

types of explicit

T(n):

(x-l)

X~

-

T(p a) log p = -~ ~(i)(0 )

0 -~-'

p <x where the sum

E P

is taken over the zeros of

~(s).

From this formula and the

results already mentioned we obtain the estimate

T(p) log p << xl3/2exp{-c(log p<__x

which in a weaker form follows from Rankin

[ 30 ]-I

x)½},

and had been hinted at in the

Princeton version of Hardy's Twelve lectures on Ramanujan tion of our explicit elliptic curve;

formulas is to the Hasse-Weil

representation

log p = ~ ~ - +

p~<_x N(p ~) = Card(EOFp~))

L(s,E)

L(s,E)

of an

is the Euler

~

- 2L(2)(0'E) " - -

+ log (x-l),

L(1)(0,E)

runs over the non-trivial

zeros of

The interest in such a formula arises from the intriguing possibility

that the multiplicity meaning

and the sum

Another applica-

then XP

(p~ + 1 - N(p~))

L(s,E).

zeta function

indeed it follows from our results that if

product of an automorphic

where

[ 13 ].

[ 5 ].

of the zero at

O = 1

does have a definite geometric

In this direction we prove that, for the Euler product L(s,~)

of a cuspidal automorphic

representation

~,

the order of vanishing

of L(s,~)

78

at the real point on the critical line, which is the point of major interest in the case of elliptic curves, is bounded from above by the logarithm of the conductor

(= level in the old terminology)

of

7.

Another problem that motivates our study of the analytic properties of the Euler products of automorphic representations, which is not entirely unrelated to the results we have stated above, is the possibility of carrying out for these Euler products the program first initiated by Riemann for die Anzahl der Primzahlen unter einer gegebenen Grosse"i

~(s)

in his "Ueber

It is to be hoped that

a closer study of Euler products from the point of view of harmonic analysis may give some information concerning Riemann's observation that Euler products only vanish on the symmetry line of the functional equation.

An initial contribution

that we make in this direction is the derivation of an explicit formula for the Euler products of automorphic representations, (automorphic)

representations

as Well

of extensions of

GL 1

[ 41 ] had done for the

of Galois type.

Our explicit

formula exhibits an element of Weyl-symmetry which closely resembles the constant term of an Eisenstein series associated to a maximal parabolic subgroup of a semisimple connected Lie group.

The content of the various sections the notation used throughout these notes.

is. as follows.

In

§2

In

§i

we introduce

we develop Rankin's convolution

trick [ 30 ]-II in a form which is most suitable to the applications we have in mind. §4

In

§3

we study the zeros of Euler products in the critical strip.

we obtain explicit formulas with error terms.

inside

the

critical strip are obtained.

In

Finally in

§5 §6

In

zero-free regions we give a derivation of

our explicit formulas with Weyl-symmetry and give also a general formulation of problems concerning the location of the zeros of Euler products of automorphic representations.

One final remark about representations

of the group

~.

In these notes we work mostly with automorphic

GL2~A) ,

where

~

is the adele ring of the rationals;

although most of our computations do not generalize to extensions of

~,

the

79

ideas do seem to apply, even to the case when linear group over a number field.

GL 2

is replaced by a reductive

We are unable to carry out such an investi-

gation at the present time due to the absence of a concept of conductor of an automorphic representation, general enough to include an explanation of the well known shape of the local factors of Artin's Euler products at the ramified primes. This we hope to pursue in future publications.

Another reason for working over

is the implications that Langlands' funtoriality principle [ 24 ]

has for the

explicit estimates we carry out in these notes and which in the setting of

GL 1

had already been hinted at by Lang [ 21].

Acknowledgment:

I am greatful to R. R. Rao, R. A. Rankin and J.-P. Serre who,

directly and indirectly, contributed many ideas developed here. received from an NSF grant.

Support was

During most of the writing of these notes the author

was visiting member of the Institute for Advanced Study at Princeton and an associate member of the Center for Advanced Study at the University of Illinois. would like to thank Nancy Lomax for her skillful typing.

C. J. Moreno February 1977.

I

80

§i.

Automorphic Representations

i.i.

and Euler Products.

Classical Automorphic Forms.

We shall follow closely the notation of Gelbart's monograph H

be the upper half plane, let

real

2 × 2

G = GL 2

and let

matrices with positive determinant"

~

= GL~)

define

all functions

g(z) = (az + b)/(cz + d)

and

f

define

on

H

and

g

in

~ ]K

Let

be the group of

for a pair

g = (c a b) E ~

'

z E H

[ i0 ].

d

and

1

j(g,z) = (cz + d)(det g)-2-. For

(fl [g]k)(z) = f(g(z))J(g, z)-k

for any integer

k.

For any positive integer

N

F0(N ) = {(a b) ~ SL2(2Z):

c _= rood" N}.

For any two positive integers space

M~(N,g)

k

and

N

(MI)

f

is holomorphic on

(M2)

fI [~]k = ~(a)-if

f

on

E

and any character

of modular forms of weight

of all complex valued functions

let

k,

H

level

N

of

and type

and observe that by g-IF (N)g

(M2)

consists

H

for all

F (N)

E

the

satisfying

~ = (~ ~) E F0(N).

The next condition deals with the regular behavior of To state it we let

(~/N) x

be the kernel of the homomorphism of fI[y] k = f

is commensurable with

for all

F0(N) ,

~ E F (N).

f

at the cusps. r0(N):

I~ ~ ) ÷ g ( d )

Now for any gEGL2(Q) ,

and hence

g-iFe(N)g A PZZ = {(~ ml): m E 2g }

for some integer

n depending on

triangular unipotent matrices).

g

(where

Since

P

is the algebraic group of upper

g-iF(N)g

fixes

fl[g] k

and

f

is

81

holomorphic we have a Laurent expansion 2 gi mz

fl[glk(Z ) = ~ am(g)e m

n

The regularity condition is then (M3) C

For every

g

GL2(Q)

A,

and positive real number

there exists a constant

such that

Ifl[g]k(x+iy)

for all

- a0(g) I j c exp(-2~y)

y > A.

The set

Sk(N,E)

of cusp forms is the subspace of functions in

%(N,E)

having the further property that

(M4)

s0(g) =

for all

1.2.

g E GL2(Q)

and

f ~/m

(fl[g]k)(z+nx)dx = 0

z E H.

Automorphic Forms on Adele Groups.

Let

~ = ~Q

denote the adele ring of the rationals and recall that the

idele group has the decomposition

/Ax

= Qx.~R+ ~

(2g )x; P

also recall that the unique factorization of an idele and the strong approximation law for

SL2(Q)

imply that

GL2(/A) = GL2(Q)'GL2@R) ~

where

Kp

is any choice of subgroups of

GL2(Q p)

Kp,

such that

Kp = GL2(2Zp)

for

82

all but a finite number of

p,

~x P

p.

is surjective for all

In our case {(: ~) E GL2(~p): divide

N.

Kp

is open for the rest and the determinant map into

will be the closure of

c E 0 mod. N}.

It is

F0(N)

GL2(~ p)

GL2(~ p)

precisely when

Koo is taken to be the orthogonal group

thought of as the discrete subgroup of

in

G/A = GL2~A)

02(~).

p

or does not

GQ = GL2(Q) is also

consisting of Q-rational

points.

The center

Z/A

(resp. ZQ)

of

GL2~A)

(resp. GL2(Q))

scalar matrices and therefore is isomorphic to the ideles

Let

p

denote the right regular representation

morphic form on

GL 2

is any function

~

on

~x

of

consists of (resp. QX).

GL2~A).

An auto-

G/A satisfying the following

conditions.

(i) (ii) (iii)

¢(yg) = ~(g)

for all

for some grossencharacter ~

is right

lates

p(z)~,

GL~ (~),

z

G

for

k E K

= GL2(~)

i.e.

for all

z ~ Z/A;

the space of functions on

is finite dimensional;

alone,

~

is smooth and the trans-

in the center of the universal enveloping algebra of

C

and

for any compact set M

Q

in

GL2~A) and

A > 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

and 2~inx gl[p]o(Z) =

Put, for

~ bp(n)y½Kr(2~K--~0) e n#0

Re(s) sufficiently large,

Lp(S,~X~')

s+~ = ~p.' -½rKP~[~]

and let ]L(s,ITxI[')

~(s,x)

F(s+~+r)F(s+~-r) F(s + ~) k

~ i ~ a (n)b (n)n s->~(k-l) n= 1 @ P

be the column vector

t(Lpl(s,~

Let

KO .

x

~') ..... Lp (s,~

x

Z')).

be the constant term matrix for the Eisenstein series

Then the vector function

~(s,~ x ~')

Ep(Z,S,X,k).

has a meromorphic continuation to the

whole s-plane and satisfies the functional equation

104

• (s,~ × ~') = ~(s,×)~(l-s,~

where

~

and

~'

are the contragredients

Remark 2.1.4. then the functional

If

rX

x

~

and

runs through the

equation for

Lp(s,TT

of

L(s,~

~T')

x ~')

x ~'),

~'

~o(N)

respectively.

rational cusps of

F (N) o

can be written also in the form

= ~ q~p,l(s,x)L%(l-s,~

x

~'),

r% where

~p,%(s,x)

is the

Remark 2.1.5. the group

F (N) o

p-%

entry in the constant

in the functional

equation for the function

S-matrix

~(s,X)

times a relatively

~(s,~

x 7')

cusp

rp

of

L (s,~ × 7') p

seems to be only of a notational

In fact there is in the literature ample evidence

that the vector valued function s

~(s,x).

The apparent scattering about each rational

which is brought about by the nature.

term matrix

([ 8 ],[ii ],[14 ],[25])

is really a scalar valued function of

simple vector valued function.

The scalar function is most

likely related to theEuler

product constructed in §i.~. for anautomorphic representa-

tion of

Below we shall give explicit examples of such Euler products

GL2~AQ) x GL2~AQ).

satisfying relatively

simple functional

In the development have, as a notational

of Rankin's

device,

equations.

convolution method it is convenient

the formal identity between fundamental

to

domains

which is given by the following lemma.

Lemma 2.3.

Let

be a rational cusp of the stabilizer

of

p

D (N) o

F (N) o be

F

p

be a fixed fundamental

and let

p E SL2(~ )

= {o E F (N): O(r) = r} o

domain for

be such that and put

F (N). o p(i ~) = r.

Let Let

r

105

where H

K

O

is a positive integer.

Define a region

S

P

in the upper half plane

by

K Ixl < ~ ,

Sp = {z = x + iy:

y _> 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


equation for the Eisenstein

series

Ep(z,s,×,k)

can be written

as

Ep(Z,S,×,k) = ~ Cp,x(s,×)kEl(z,l-s,~,k),

where

~

runs over a complete set of inequivalent

this into the integral representation

of

cusps of

Lp(S,%X~')

We substitute

F (N). o

to obtain

Lp(S,%X%') = Z ~p,%(s,X)kLl(l-s,#X#'),

where ~'

~

and

are the automorphic

representations

which are paired by the correspondence

now denotes and

~'

if

Sl + ~ i

the vector function whose components

~(s, X)

is the constant

contragredient

and

g2 ÷ ~2"

are the functions

If

to

~

and

l(s,~X~')

L (s,~X~')

term matrix then we have

]L(s,~TX~')

= C>(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 <

i, and choose once and for all a n.

set of integers

{d I .... ,d%}

such that

d. ~ 0 mod. Ni, 1

d. ~ 1 mod. pi l, ] < i < 1

% The mapping

~

÷ ~

g i v e n by X {a I ..... a%} ÷ a =

~ d.a. i 1 i=l

A induces

a ring isomorphism

of

2Z/N

onto

ni

T7 z ~ / ( P i )" i=l

Under

this mapping

we

have

(a(1),a(2),N)

= i

if and only if

(a (I) (2) i 'ai ,wi) = i

and

(u,N) = 1

Let

I = llX...xl%,

We identify V(1)

I

if and only if

where

(ui,Pi)

I i = {(al,a 2)_ mod.

with a subset of

be the space of functions

(ZZ/N)X(TZ/N) on

I.

= i,

1 < i < %.

ni Pi : a l = i or a 2 = i, a I - 0 mod. via the above

isomorphism.

Let

We then have

V(I) = V(II)@...@V(I%).

For each character

X mod.

N,

there exists characters

)
n. Pi I

such that

t X(a) = ~ xi(ai). i=l

Let

r i = r(xi)

+ 0 if

i < i < ~

there exists a primitive character

and r.l = 0 P r.

X mod. " ~ p i 1 i=l

X(a) = x(a)

if

if

such t h a t

(a,N) = i.

D+I < i < ~.

Then

pi }.

110

If

T.

is a linear transformation

on

V(I.)

i

for

i < i < %

then

T = TI@...®T %

I

induces a linear transformation

on

V(1).

al s = a--

associate the rational cusp

of

To each element Fo(N),

s = {al,a 2}

in

I

we

and let

2 aI * p = (a2 ,) ~ SL2(2Z)

be a representative the stabilizer

of

al p(i~) = - - . a2

element with s

in

F (N)

It can easily be verified

that

is

O

2 aln

l-ala2n

Fp = {(-a~n

.. N

l+ala2n):

n ~ ~ • (a~,N)};

clearly

F

is the stabilizer of is

(a~,N).

correspond

p

s

nF

in

=

N

{(l-ala2n -a~n

FN

2 aln ) n E ~ "N} l+ala2n :

We now consider a distinguished to each element

Ep(Z,s,x,k ) =

[Fp: Fp A F N]

and the index

s E I

set

of

the Eisenstein

functions

put in place of

If

p

on the element the symbol

r = {a,b}

V(I)

in

Fp

by letting

~ X(O) fj(p-lO'z) )k(imp-ld(z))½(s+l) de ro(N)/rp ~lj(£1d,z)I

imaginary axis the axis of symmetry. Ep(z,s,x,k)

in

F P ~ FN

series

Notice should be made of the natural change of variable

series

of

and

s+l s + T

which makes the

To indicate the dependence of the Eisenstein {al,a 2}

of

I

we will find it convenient

'*'

r' = {a',b'}

are elements

a


= det.

a T

(b b ')"

in

I

we put

to

111

The main result of Orihara ([ 28 ], p. 141) can now be stated in a slightly generalized form as follows:

Theorem 2.5. series

Ep(z,s,x,k )

With the notations as above we have that the Eisenstein satisfies the functional equation

[F,,F, n rN]E,(z,s,x,k ) k --

(_i) 2

s+l.

r (-~--)

A<s,~) A(s+I,~)

-.s-k+l. r (.s+k+l ~) ~ ~--T--)

9(i)®'" .~(%~F,,F ' n

FN]E*(z'-s'x'k)

where

^(s,X) : ~-½(s+A)r(S--~l)~ (l-x(p)p-S) -I P

with

A = 1

if

X(-I) = -i

linear transformation of

~(i)(a,b)

and

V(I i)

% = 0

if

X(-I),

and for

i ~ D,

~(i)

is the

with matrix entries defined by

: -- ((a,b>, -sk-ni+k Xi ~ ) P i

if

ki p II a-b, 0 < k < n.-r. 1 1

Pi 0

and for

i > ~

otherwise

the linear transformation

s

~(i)(a,b ) = (X(Pi)Pi)

l-n i

~(i)

has matrix entries

Pi - 1 • ~( ) s+l X Pi Pi - 1

if

a

=

b

k-n.+l ~(Pi)P~ - l = Pi

Remark 2.2.1. Kubota ([ 17 ],P. 69)

l

~, ) s+l " (~(pi)p~) -k ktPi Pi - i

if

P~II a-b, O
The proof of this result is identical to that given by for a similar result.

To make use of the ideas of Orihara

112 ([28 ], p. 141),

one simply observes

that for a suitable class of functions

defined on the upper half plane and a Dirichlet one has the simple combinatorial

aeFo(N)/rp

where

X(°)fo(z) =

X(o) = x(d) if

the constant

[Fp

Fp

:

1

[

l" n

P

d = (, d ).

and (2.5).

N

[

treated by

term matrix for the Eisenstein

Let the assumptions

Then the Euler product

to realize that

s+l E(Z,T ,x,{al,a2} ) .

by the principle expounded

fu~(Z),

OeFN/F p n FN

It is then not very difficult

in Kubota

We can now state a more precise form of Theorem

Theorem 2.6.

defined modulo

×(u)

PeFo(N)/F N

FN]

term matrix for the Eisensten

n FN]Ep(Z , s,x,k)

X

identity

Orihara is related to the constant [rp:

character

fo(z)

series

[ 17 ], p. 69.

(2.4).

and notation be as in Theorems

Lp(S,~×~')

satisfies

(2.4)

the functional

equation

l+s [F,: F, n FN]L,(--~--,~×~') k (-l)2F(S--~l) 2 _.s+k+l.~.s-k+l.

Proof.

A(s,~) }(1)®...@}G)[F,: r, n FN]L,(~,{ x~') A(s+l,b

From the proof of Theorem

(2.4)

we recall the integral represen-

ration k .l+s Lp[T,~x~')

=

f

y2f(z)g(z)Ep(Z,S,7,k)dQ.

Do(N)

Observe again that we have made a natural change of complex variable the definition of the Eisenstein

series.

s +

s+l

We now multiply both sides of this

in

113

equality by the factor the functional

[Fp: Fp n FN] ,

equation for

replace the symbol

[F,: F, N FN]E,(z,s,x,k )

O

by

'*'

and apply

given in Theorem

(2.5).

This proves the Theorem.

Remark 2.2.2. s = 1

if and only if

under consideration

We have already remarked X

is the principal

Lp(S,~×~')

that

Ep(Z,S,X,k )

character and

has no pole at

s = i,

k = 0.

has a pole at In the case = @~

since

and P

7' = 8~' P

have non-equivalent

components

if in the notation of Theorem

at the infinite prime.

More generally,

(2.4),

y½(l+s) + Cp(S,X)y½(l-s)

is the constant term of

Ep(z,s,x,k)

about the cusp

residues on both sides of the integral representation

p,

then a comparison of

for

.l+s L[-~--, 7 x ~')

gives

that

.l+s Residue L £ T , ~ × ~ ' ) s=l

where

< f,g)

associated

to

2 automorphic Euler product

is the Petersson ~

and

7'.

inner product of the automorphic

forms

f

This identity which relates the equivalence

representations L(s,~ × ~')

= (f,g) Residue ~p(S,X), s=l

7

at

and

s = 1

~'

and

g

of the

with the analytic behavior of the

will play a fundamental

role in the investi-

gations to follow.

Theorems and

~'

(2.4)

and

which are allowed

consequently

(2.6)

dealt with automorphic

to have a finite number of ramified local components;

the functional

contained rather complicated

equation satisfied by the Euler product factors corresponding

simple functional

equation.

Lp(S,~ x~')

to the ramified components.

In contrast with this one can prove that in the unramified satisfies a relatively

representations

situation

L(s,7×7')

114

Theorem 2.7.

Let

associated respectively and

g(z)

in

~

and

~'

be automorphic representations

to real analytic wave forms

WI(I, %') ,

%' = l-r' 2 '

i.e.

f(z) in

WI(I'%)'

of

GL2(/A)

l-r % = T

real analytic forms of level

i

which are eigen functions of the Hecke operators.

Suppose the Fourier expansions

of

at infinity are given by

f(z)

and

g(z)

about the cusp of

F (i) = F o

f(z) = n~0~ a(n)y ½ Kr(2~ Inly)e 2~inx

and

½ inly ) 2~inx b(n)y Kr,(2~ e

g(z) = n 0

Suppose that

f(z)

and

g(z)

satisfy

(f(z)IK)(g(z)IK)

where

K

is the operator

L(s,~X~')

= -s~

= f(z)g(z),

g(z) IK - g(-z).

Define for

(s+r+r' s+r-r' (s-r+r'] (s-r-r' ~ ) F ( ~ ) F . ~ . F . ~ )

Re(s)

large

~a(n)b(n) n-s-

4F(s)

Then we have

.l+s n[ 7 ,Tx~')

A(s) L ( ~ , w ~ A(s+l)

x w'),

where

A(s) = ~-½sr(~)~(s).

Furthermore s = I

L(s,~XT')

is regular in

which occurs if and only if

precisely we have

Re(s) > ½

except for a simple pole at

is the contragredient

of

~'.

More

115 co

= ~3( f , g > "

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
then we have

L(s,~X~') =

The functional equation for A(s) = A(l-s)

L(s,~ ×~')

L(s,~,~';r@r) 4A(2s)

together with the functional equation

now gives the result

L(s,~,~';rSr) = L(l-s,~,~';r~r).

Furthermore, since

L(s,~,~';r@r)

differs from

L(s,~X~')

by the factor

A(2s),

120

it is also holomorphic in the region if and only if

v

Re(s) > ½

is the contragredient of

and has a simple pole at

~'

or equivalently if

s = 1

z = z[

It

remains an open question to investigate the location of the zeros (and poles) of the Euler product

L(s,~,~';r®r).

Example 2.

Let

~

and

~'

be automorphic representations of

associated respectively to holomorphic cusp forms of weight

k

of level i.

and

If we put

% =

k-i

--~-'

then the Euler products of

~

and

%,

k-i

%

=-~-'

~'

i-i o

and

£

GL2~A Q) and both

%'

2'

~+i

o=--7

-,

have the form

-h(s-%)~.s-L -½(s-%')~s-%'.v-r._ _ -s.-l~. =- -s,-i L(s,~) = Tf I~--~-)7T y~---) 11 [±-%pp ) [--%pp ) P

=

L(I-s,~)

and

L(s,~') =

-½(S-%o)F(~)-½(S-%o)F(~)77-

(l_~;p-S)-i ( l - ~-~ p p ) -s.-i . P

If we define a F-factor by

4 i=l

where

~+__~k %1

=

1 -

2

' %2

2

%-k

Z-k

~+k =

' %3

=

2

' %4

=

2

-

i,

and if we put

L(s,~,~';r@r') = g ( s , ~ , ~ ' ) ~ d e t P where

{gp}

(resp.

{g;})

(14-p-sr(gp)®r(gp))-l,

is the conjugacy class in

GL2(~)

containing the

121

0

matrix

(0P ~ o ) ( r e s p .

tation of

GL2(~) ,

o

( P ~.)), and

r

is the standard 2-dimensional represen-

then we have by a simple application of Rankin's convolution

method

L(s,~,~';r@r)

where

= L(l-s,~,~';r@r)"

(-l)mF(s) 2 F(s+m)F(s-m)'

m = k - %.

Remark 2.3.1.

It should be observed that the extra

F-factor appearing

in the functional equation is a simple rational function of s.

Again from the

properties of the Eisenstein series we have that the Euler product is holomorphic for s = 1

Re(s) > 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) ' <0 f~q~ ~'-~0~q~q]

and the products (NI,N2),

N2

~',

but not

~", N1

~"', and

run respectively over the primes that divide

N1

L(s,7,7';r®r)

Furthermore,

L(s,~,~';r @ r)

simple pole at

s = 1

if

but not

= M

N 2.

~--SL(l-s,7,~ ;r®r).

is holomorphic for

7

We then have

Re(s) > ½

is the contragredient of

the equivalence of the local components of

7 = ~7

with a possible

7'.

and

The requirement about

7' = 57' P

P

at the infinite

prime can be relaxed in the proof of the above functional equation by making some trivial changes in the argument in [ 25 ].

Example 4.

The following example, whose significance is best understood

in the contest of the theory of automorphic representations for obtained by Shimura ([ 38 ]).

GL3~A),

was

The method of proof generalizes in a non-trivial

way Rankin's convolution method to automorphic forms on the Metaplectic group. Let

7 = @~Tp be an automorphic representation of

morphic cusp form of weight

k

and level

I.

7 . P

2-dimensional representation of

r2

We define a

and

Let

= ~ 7-½(s-li) F ( ~ ) i=l

with

l I = 0, ~2 = -i,

and if we put

rI

p

let

{gp}

13 = k - 2

be

be the standard

the symmetric square of

F-factor by

F(s'7'r2)

associated to a holo-

For each prime

the semisimple conjugaey class associated to GL2(¢)

GL2~A ~)

r I.

124

L(s,~,r2) = r(s,~,r2)~det(13-p-Sr2(gp)) -I, P then Shimura's result, which is valid for a more general automorphic representation 7,

is that

L(s,~,r2) = L(l-s,7,~2)

and, most important of all, proved that tation of

L(s,~,r) GL3~A~) ,

Example 5.

L(s,~,rT)

is entire.

Gelbart and Jacquet [ ii ]

is in fact the Euler product of an automorphic represenin accordance with

Langlands' functoriality principle.

This example, due to Doi and Naganuma [ 8 ],

of the earliest evidence of how an automorphic representation could be lifted to an automorphic representation of quadratic field.

have

Let

7 = ~p

GL2~A F)

~

provided some of

with

GL2(/A~) F

be an automorphic representation of

a real GL2~A Q)

whose component at the infinite prime belongs to the holomorphic discrete series and is unramified at all local components, i.e. cusp form of weight of

GL2~A~)

k

and level

i.

Let

7'

7

is associated to a holomorphic

be an automorphic representation

associated to a real analytic wave form in

Let the Euler products of

~

and

7'

Wg(N,%),

with

~ =

l-r 2 "

be respectively

L(s,~) = r(s,z,r)~det(12-p-Sr(gp)) -I P and

L(s,~') = r(s,z',r)

where and by

~q r

q~N (l-6qq-S)-ip~N det(12-P

is a complex number and {gp}, {g~}

r(gp))-i

are conjugacy classes in

is the standard 2-dimensional representation of

GL2(~).

GL2(K)

Define a r-factor

125

F(s,~,~';r@r)

= ~~-]H~-½(s-ki)F(~ii) i=l

with

k-I Ii =

k+l

2

r, 12

2

k-I r, ~3 =

2

k+l

+ r, ~4 = - - - +2

r~

and put

L(s,~,~';r®r)

=

-i - q-Sr(~qgq))-ip~Ndet(l 4 - p-Sr(gp) ® r(gp) )

= F ( s , z , ~ ' ; r @ r ) ~2d eNqt (ll

We then have that

L(s,~';r®r)

L(s,~,~';r@r)

Here

W(~,~')

= W(~,~')N2(½-S)L(l-s,{,@';r@r).

is a complex number of absolute value 1

subsequent discussion. 7'

is holomorphic and satisfies

which plays no role in our

To obtain the result of Doi and Naganuma, one takes for

an automorphic representation associated to a 2 dimensional semisimple repre-

sentation of the Weil group of a real quadratic extension

F

of the rationals

which is induced by an unramified grossencharacter of the idele class group of

F.

Example 6.

The following example was first pointed out by Langlands and

was suggested by his theory of Eisenstein series.

Due to our lack of information

about ramification phenomena, we consider only the simplest situation corresponding to an automorphic representation

~ = ~

of

GL2~A ~)

of conductor 1

associated

P to a holomorphic cusp form of weight k. standard representation of in

GL2(~)

GL2(~).

Let

Let

r3

{gp}

attached to the local component

be the symmetric cube of the

be the semisimple conjugacy class

~ . p

Define a F-factor by

126

= ~

F(s,~,r3)

~-;~(s'li) F (q--), s-li

i=1

where 4

det(l 4 - r3(X )T) = ~ (i - %iT), i=l

and

{Xoo}

is the semisimple

conjugacy class corresponding

~ .

Define the Euler product of the Data {~,r 3}

as in

to the local component

§1.4

-s

L(s,~,r3)

= F(s,~,r3)~det(l P

4 - p

by

-i

r3(gp))

•

We then have

L(s,~,r3)

where the constant g(~,r3)

is independent

The proof of this functional the examination

= 8(~,r3)L(l-s,~,r3) ,

Example 7.

s

and plays no role in our results.

equation is given in Shahidi

of a 'non-constant

exceptional Lie group

of

term'

Euler products obtained b ~ base change.

consider only an automorphic to a holomorphic

G

~

of conductor

ard 2-dimensional the Frobenius class in

and let f(o)

and let

representation

associated

jugacy class of a generator K ~ . given by

~ = 97

cusp form of weight

G = GaI(K/~).

Let

X = trace 0-

P

k.

[ 22 ]

of Let

0: G + GL(V) Let

For simplicity we GL2~A~)~ K

be a representation

GL2(~).

For a finite prime

G

and let

{gp}

Gal(K ~R) w

where

Recall that the r-factor associated

of conductor i

be a finite Galois

of

p

be the semisimple

K

~ . P w

con-

of the Saito-

r: GL2(~) ÷ GL(W)

to the local component of

The following

representations.

representation

eonjugacy class in

GL2(~)

series of the

G 2.

Shintani theory of liftings of automorphic

extension of

and depends on

of a certain Eisenstein

jectural example is suggested by Langlands'reformulation

associated

[ 36 ]

Let

{~ }

of

be the standlet

~

P

be

conjugacy be the con-

is a simple factor of

to the Artin L-function L(s,O)

is

127

dimo

~.

~

,,

•

.

r(s,~) =7T ~-~ts-~r(~) i=l

with

hi =

Recall that the

0

if

I < i < X(1)+X(O°°)

-i

if

)<(1)+X(°°°) < i -< dimo. 2

F-factor of the Euler product

F(s,~,r)

2 =~j=l

L(s,~,r)

is

~ ~o, ~o 2(s-aj) r(s-!J) 2

with

o = k-i o hl - --2-' %2

We define a new F-factor associated

r(s,~;r®p)

k+l 2 "

to the data

{~,r,p)

by

2 dimp h . .o ~ , 4o = ~T ~ 7-21S-Aj--Ai) r(S-A$--Ai). j=l i=l 2

If we put

L(s,~;r®p)

= £(s,~;r@p)~det(I

- p-Sr(gp)@plVIp(Op))-l,

P

where

VIp

is the subspace of invariants

of

V

of the inertia group

Ip, then

it is to be expected that

L(s,z;r@p)

where dent of

~

= £(~,r®p)f(0)2(½-S)L(l-s,~;r®P),

is the contragredient s

of

and of absolute value i.

representation

of

G

and

p

and

g(~,rSp)

is a 'root number'

If this is true and if

PK

indepen-

is the regular

128

PK =

is its natural decomposition

[~ (dim@)~ ~eG

then the Euler product

L(s,~;r®PK) = ~^L(s,~;r®@) dim@ ~eG would satisfy the functional

equation

L(s,~;r@PK ) = £K(~,r)d~(!~-S)L(l-s,~;~®~K),

where

dK

is the absolute value of the discriminant

functoriality automorphic

principle

suggests

representation

of

that

GL2~)

Shintany theory as given in [22 ] If one is only interested,

L(s,n;r®p K) ;

K/~.

as we are in these notes,

is 2-dimensional

of the Euler product

L(s,~;r~PK),

decisive

representation

get the analytic continuation representations necessarily

then the Saito-

information

~

L(s,~;r®p)

n(p)

of

of the Euler product

9-

The simplest

Using Rankin's

and in some cases the

L(s,p)

GL2~A~).

such situation

that

is the Euler

This allows us to

L(s,~;r@O K)

which may be ramified and number fields

abelian over

about the

under the assumption

and its associated Artin L-function

product of an automorphic

Q.

in the analytic continuation

convolution method we can obtain the analytic continuation,

p

k is cyclic over

nature of the Euler product for abelian extensions.

equation,

Langlands'

is the Euler product of an

in the special case when

Shintani theory also proves this and provides

functional

In fact

this is indeed proved by the Saito-

and the functional equation of the Euler product

holomorphic

of

K

for automorphic which are not

arises when

K

is the

splitting field of a cubic polynomial with integer coefficients

whose discriminant

is not a square,

group

in which case

whose representations

Example 8.

GaI(K/~)

is the full symmetric

S 3,

all of

are monomial.

An Euler product that contains

two variables.

Let z

be an

129

unramified automorphic representation of analytic cusp form

f(z).

GL2~A ~)

which is associated to a real

Let

f(z) =

[ a(n)y2K%(2~Inly)exp(2~inx)

n#O be its Fourier expansion about the cusp at infinity with

a(1) = i.

Let

E(s,z)

be the Eisenstein series of

§2.1.

functions

o (n) s

and Rankin's convolution method show that for complex

variables

s

and

and

a(n)

s'

The multiplicative property of the arithmetical

the expression

L ( s ' , s ; ~ E)

=

/

E(z,s)f(z)S(z,s')d~

Do(F) -½(s'+l)

s ~2s+l

r

s 4J~s-I

s --~s+~

(----if----) r (---y--) r ( T )

s - 2s+~

s --~s-I

4r (s') A (s+l)

is an Euler product.

n~oOs~Inl) I t? a-~)

The functional equation for the Eisenstein series

gives readily the following functional equation for

..l+s' c~--y-,s;~ E)

=

A(s')A(s) A(s'+l)A(s+l)

In fact a direct computation shows that if automorphic representation L ( l+s'+s ~ , ~ ) L ( - - - --l+s'-s ~--,~)

Example 9.

~-(s,+½s)

r (---y--) r (-----if----)

7,

then

E(z,s)

L(s',s;~®~ E)

l-s' L(--~-,-s;~e~E)

L(s,~)

is the Euler product of the

..l+s' L[--~--,S;~T E)

is actually equal to

times a simple expression that involves only A(s).

Let

S

be the symplectic group of rank 2

and let

~ = ~

P4

P

be an automorphic representation of components.

For a finite prime

in the dual group

S05(E).

p

Sp4~A ~) let

{gp}

which is unramified at all the local be the semisimple conjugacy class

Assume that the component

~

belongs to the

130

holomorphic

discrete series of 'weight'

modular form of weight Hecke operators. S05(~).

Let

k

k,

that is

~

is generated by a Siegel

which is a cusp form and an eigenfunction

r

be the natural

4-dimensional

for all the

complex representation

of

Define a F-factor by

4 = ~-½(s-%i)r(s-%i), i=l 2

r(s,~,r)

with

~I = k-i 2'

~2 - k-i 2

i, %3 = -%1' ~4 = -~2"

If we put

L(s,n,r)

then Andrianov

[ 1 ]

= F(s,~,r)~det(l P

has shown that

L(s,n,r)

It is also shown that be observed

that

4 - p-Sr(g ) ) - i P

L(s,~,r)

L(½,~,r)

=0

= (-l)kL(l-s,~,r).

has at most a finite number of poles. for

occurs for classical automorphic

k

an odd integer,

forms on the group

It should

a phenomenon which already

Sp2

=

SL 2

of weight

k s 2

mod 4.

2.4.

The Average Size of the Eigenvalues

of Hecke Operators.

In this section we obtain information that appear in the Fourier expansions be used to determine L(s,~);

of automorphic

forms on

GL2~A ~)

which can

the location of the critical strip for the Euler products

we also get some information

correspond

about the size of the coefficients

to the unramified

about the zonal spherical

local components

~

P

functions

of an automorphic

that

representation

131

= ~

. We first prove a simple result that suggests that the Petersson-Ramanujan P conjecture is true on the average for real analytic cusp forms.

Theorem 2.8.

Let

~

be an automorphic representation of

is associated to a real analytic wave form

f(z)

in

GL2~A ~) which

Wx(N,%), % = ½(l-r2).

Let

its corresponding zeta be

oo

~(S,Z) =

[ a(n)n -s n=l

= q~N(l-a (q) q-S)-ip~N(l-a (p) p-S+x(p) p-2S)-i.

Then we have for

x

a positive real number

la(n) I2 = c(~)x + o(x), n<x

where

3 1 c(Z) = ~'F(½+r)F(½ r ) ' N ~ ( l + p

i)"

pIN Proof. of

Fo(N)

Let the Fourier expansion of

f(z)

about the cusp at infinity

be

f(z) =

~ a(m)y=Kr(2~Imly)exp(2~imx). m#0

The modified Bessel function

I

Ks(Z) = 2 "

where

-S

(z) - Is(Z) '

sin(s~)

132

(½z) s+2m

I(z) = m ~ 0 ~ l )

is clearly real valued when valued and

s

and

s is pure imaginary.

z

are both real and also when

z

is real

First we note that ~½

/ / If (z) I2ySd~ o -½ oo ½ -~ ½ = f yS a(m)y Kr(2~Imly) ~ a(n)yaKr(2~Inly) / e2~ix(m-n)dx'y-2dy m#0 n#0 -½ o co eo = 2 ~ la(m)I 2 /oKr(2~my)Kr(2~my)yS-ldy. m=l

Again we use the identity oo / K~(~t)Kv(~t) t-Odt o

aP-i

l-p+B+v

2P+2F(I-p)

with

~ = 2~n,

p = 1 - s

l-p+p-v

l-p-p+v

and

p = ~ = r

to get

~½

/ f If( z ) 12ySd~ o -½ (4~)-sF(~)F(~)F(2)2 4F(s)

co~ la(n) 12n-s n=l

= L(S,~ x~).

Recall that if

l-p-~-v

.r(----y--)C(----F--)r(~)r(----f----)

Foo = {(i0 i): n ~ 7z}, S

S = {z = x + i y :

is the region

Ixl __<½, y > 0}

133

and

D (N) o

is a fundamental domain for

r (N) o

then

S E Z ~Do(N),

where modulo

the sum r .

E

runs over a complete set of coset representatives

We can therefore

integral expression for

replace the region

L(s,~X~)

L(s,~X~)

S

If(z) 12Y sd~

o ao (N)

o

= ~ / If(z) 12ySd~°o. 0 D (N) o

The automorphy property of

f(z)

f.az+b. [c--z-~) = x(d)f(z)'

and the

SL2-invariance

of the measure

(a b

e d ) C Fo(N )

d~ = y-2dxdy

imply

If(z) I2ySd~o~ = If(z) 12(Im~(z))Sd~.

This then gives that

L(s,~X~)

= / If(z)[2E(z,s)d~, D (N) o

where

S(z,s)

= [amo(~))

O

= ySFN(Z,S )

s

r (N) o

by its equivalent in the

to obtain

= ~ I

of

is the Eisenstein series for the group infinity.

Fo(N)

corresponding to the cusp at

We now investigate the constant term of

observe that if

F

E(z,s)

more closely.

First

is the stability group of the cusp at infinity then an

equality of cosets

a b

a'

b'

F~( c d ) = F (c, d , )

occurs if and only if

(c,d) = ±(c' d'),

so the left cosets of

in i-I correspondence with the pairs (c,d); atives the pairs

(0,i)

and

(c,d)

with

hence we may choose c > 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 << x. n<x

To obtain estimates which are more precise than those of Theorem (2.8) one must use the full strength of the functional equation for this we proceed to do now.

Theorem 2.10.

(I)

Suppose the following six conditions are satisfied:

BI,...,~4, 61 .... ,64

are positive real numbers and

~i,...,~4, yl,...,y4

= Y1 + "'" + Y4 - ~i - "'" - ~4 > 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<_x

c(k)p(h)

Z kh2<x

=

~ h<x½ -

-

>(h)

~ c(k) k<-x --h2

h<x½

h

c (7) x h<x ~

3

h

but

Co (~)Xh!x~~h)h2 : Co<~)x{~ + 0(x%} and the error term gives 3

o( x5

6

3

Z ½h5) = 0(x5) • h<x

The final result is

Theorem 2.11. have that

With the notation and assumptions as in theorem 2.8., we

143

3 [ la(n)1 2 = c(T[)x + 0(x~). n<x

If in the above theorem we replace

x

by

N

and

N-I

and subtract

the

two expressions we find that 3 la(N) 12 << N~

or

3 la(N) l << N I0,

where the implied constant depends on the automorphic state this result in group theoretic

Theorem 2.12. with

z

Let

w = @~T P

a member of the principal

for which

~

be an automorphic series.

Let us

representation

of

GL2~A ¢)

Then for all those finite primes

is a class one representation

~

z.

terms.

either

P series or

representation

~

belongs

p

to the principal

P belongs

to the complementary

series

~(~i,~2)

with grossenchar-

P acters

p1

and

P2

g i v e n by

h(x) = Ixl%

and

h(x) = ,~,-°

3 0 < o < TO" 3 Proof.

la(N)[ << N I°

From the estimate

1

=

1 - a(p)T + X(p)T 2

has radius of convergence

r

with

r J~0"

it follows that the power series

~ a(pn)Tn n=0

Therefore by Gelbart

([ i0 ] p. 72)

144

3 la(p) l = Ip°P + p-°P I <__ 2pI--~

and hence the estimate

0<(7 --

Remark 2.4.1. that if

~ = ~p

3 <-i0"

p

--

It has been conjectured

is an automorphic

a finite number primes

p

([12 ], P. 357 and [33 ], P- 264)

representation

the local components

of

z

GL2~A~)

then for all but

belong to the principal P

series. = ST

A more comprehensive is an automorphic

conjecture

of Langlands

representation

of

~A'

([24 ], p. 56)

is that if

then for all primes

p

the

P character of the representation

~

is a tempered distribution.

If

~ = ~

P is an automorphic

representation

morphic discrete series, ture establishes

of

P GL2~A~)

with

z

a member of the holo-

then Deligne's proof of the Petersson-Ramanujan

this; if

z

is not amember

of the holomorphic

then the above results suggest that this conjecture

discrete

conjecseries,

is indeed true, at least on

the average.

The earlier results of Rankin of the coefficients given by Theorem

of a holomorphic

(2.8)

Theorem 2.12. and let

L(s,Z)

Let

about the average size

cusp form of arbitrary

level and the estimate

imply the following result. Let

z

be an automorphic

be its associated Euler product.

located in the strip

Proof.

([ 30 ], p. 357)

0 < Re(s) < i,

z = ~

i.e.

representation

GL2~A~)

Then the zeros of

the critical strip is

and disregard

of

L(s,Z)

are

0 < Re(s) < i.

the finite number of local represen-

P tations which are not class one and construct the remaining

local factors.

the modified

zeta

~(s,Z)

with

Then

co

1 ~(s,Z)

~'(i p

- a(p)p -s + X(p)p -2s) =

[ A(n)gz(n)n n=l

s,

where the prime in the product denotes that only those local factors enter which arise from class one representations. of the coefficients

a(n)

and

A(n)

By Theorem ( 2 . 8 )

and the multiplicativity

we obtain easily the estimate,

for large

x,

145

IA(n)~(n)l

<<

x.

n<x

Hence the series and also the product converge absolutely this implies that

~(s,~) -I

connot happen unless

represents

~(s,~)

the Euler product itself one uses the properties

with

known estimates for

q

of the

does not vanish in

dividing

([25 ], p. 295)

= ~

is an automorphic

equation.

§i.

The

for these one uses the

representations series.

L(s,~)

of

GL2~A ~)

have no zeros on the lines

with the following

representation

of

is

Ina latter section we

This will in turn lead to asymptotic

We end

Re(s) < 0

all have the form

to those that are common in the study of the distribution

Remark 2.4.3.

and

that the location of the critical

of Eisenstein

shall prove that in fact the Euler products Re(s) = O.

~;

To see that

This proves the theorem.

It is indeed remarkable

governed by the general properties

and

~(s,n)

this

to show that these local factors do

strip for the Euler products of automorphic

Re(s) = 1

in

the conductor of

not vanish outside the critical strip.

Remark 2.4.2.

Re(s) > 1

F-function and the functional

and

Re(s) > if

does not vanish in the same region.

L(s,~)

a(q)

Re(s) > 1

a regular function in

finite number of local factors not considered (i - a(q)q -s)

for

GL2~A ~)

estimates

similar

of prime numbers.

important observation. and if its associted

If

zeta

P ~(s,~),

that is the Euler product of

~

without

the

F-factors,

has a Dirichlet

series expansion of the form

~(s,~) =

~ a(n)n -s n=l

and if all but a finite number of the local components series then for large

x

la(n) l << x, n<x

belong to the principal P

146

w h e r e the implied constant is independent of stant w i l l w o r k for

all

7.

(2.8)

in the sense that the same con-

This is indeed the case for a

a m e m b e r of the h o l o m o r p h i c discrete series. from Theorem

~

~ = 8z

p

with

In general the estimates o b t a i n e d

only give the b o u n d

]a(n) l << ll~llx, n<x

w h e r e the inplied constant is independent of w h i e h m e a s u r e s the global size metric

of

~

but w h e r e

II~II

is a constant

% and is a constant m u l t i p l e of the P e t e r s s o n

< f,f > of a n o r m a l i z e d a u t o m o r p h i c form a s s o c i a t e d to

~.

All of our

subsequent estimates w i l l be effective only in so far as we are w i l l i n g to carry the constant representation

II~II ~.

as an independent parameter among those that c h a r a c t e r i z e the

147

§3.

3.1.

Zeros in the Critical Strip.

The Hadamard Product Formula.

We begin now the study of the distribution of zeros in the critical strip of the Euler products GL2~A~).

L(s,~)

associated to automorphic representations

The methods, worked out here only for GL2,

~

of

are quite general and apply

to the Euler products of other adele groups over algebraic number fields.

In this

more general setting we have investigated some analogues of the Brauer-Siegel Theorem

in the theory of Artin's L-functions.

Throughout this section we fix an automorphic representation GL2~A ~)

and denote, as usual, by

L(s,~)

its Euler product.

Hadamard factorization theory to the entire function that there exist positive constants

c

and

c'

L(s,~)

such that

~

of

To apply the we must first verify

L(s,~)

satisfies the

growth condition

(1)

for

(2)

L(s,~) = O(exp cls I log Isl)

Isl

sufficiently large, but the weaker condition

L(s,~) = 0(exp c'Is I)

does not hold for all sufficiently large

Isl.

We first decompose the Euler product

L(s,Z)

into its

F-factor and the

product of all local factors at the finite primes:

L(s,~)

To investigate the growth behavior of

= r(s,~

~(s,~)

)~(s,~).

we consider its associated Lindel~f

148

function

p(O,z).

of numbers

~

Recall that for each

0

p(O,z)

is defined as the lower b o u n d

such that

~(~ + it,~)

o(ItI$).

It is w e l l k n o w n f r o m the general theory of Dirichlet series that, as a f u n c t i o n of

~,

~(~,~)

is continuous,

that no arc of the curve

non-increasing,

y = ~(~,~)

and convex downward in the sense

has any point above its cord; also

B(~,~)

is never negative.

We have a l r e a d y indicated in

§2.4

that

¢(s,~)

gent, as an Euler product or as a Dirichlet series, ~(s,~)

is b o u n d e d for

o > 1 + 6(6 > 0),

p(o,~)

for

is a b s o l u t e l y converRe(s)

> i;

in p a r t i c u l a r

and therefore

= O, ~ > i;

from the estimate

r(l-s,~) F(s,~)

where

= max(I~lI,I~21),

0((It I +

I~l) 2(~-½))

and the functional e q u a t i o n of

L(s,~)

it follows

that

p(o,~)

= 2(½

- 5),

by c o n t i n u i t y these equations also hold for

The cord j o i n i n g the points is

y = i - ~;

(0,i)

o < 0;

o = i

and

and

(i,0)

~ = 0

on the curve

from the convexity p r o p e r t y it then follows that

p(o,~) < i - ~,

0 < o < i,

respectively.

y = ~(~,~)

149

and in particular

~(½,~) J ½, that is

~(½ + it,~) = O([t[ ½+£)

for every positive between D(O,~)

0

and

i.

consists of

~.

The exact value of

is not known for any

2

straight lines:

More generally,

if

for

~

~ < ½

and

~(o,~) = 0

Hypothesis for the factor

for

o > ½.

is an automorphic representation of

is a finite dimensional complex representation of

§1.4

o

An analogue of the Lindelof Hypothesis is that the graph of

~(o,~) = (½ - 4)2

r

~(~,~)

~(s,~,r)

GL2(~)

in the Euler product

GL2~A ~)

and

then the Lindelof

L(s,~,r)

defined in

is that the graph of the associated Lindelof function consists of

2 straight

lines:

~(o,~,r) = (½ - o)dim r

for

O < ½

and

~(O,~,r) = 0

for

~ > ½.

By Stirling's formula we know that

F(s,~) = 0(exp c[s] log Is]), Re(s) ~ ½,

with F(s,~)

c

a positive constant. and

~(s,~)

The function equation and the above estimates for

give

IL<s,~)l = O<e=p clsl ~og I s l ) .

This proves

(i).

If we now restrict

s

to the real axis and put

s = ~ > ~ --

then O

150

Stirting's

formula gives

I£(s,~)l ~ exp aO log o

for some positive constant

a.

Fourier-Bessel

of the automorphic

coefficients

On the other hand the polynomial form associated

growth of the

to

~

imply that

oo

l~(O,v) l = I ~ a(n)n-Sl n=l co

> 1 -

Z la(n)In -~ n=2

>½

for

o

sufficiently

say

o _> Go,

large.

Therefore,

for large real positive values of

~,

we have

IL(o,~) I > exp c'O log o,

where

c'

is a positive constant.

We have thus verified Hadamard Factorization conclusions

that

L(s,~)

(2)

cannot hold for all

satisfies all the requirements

Isl.

of the

for subsequent use we now collect the resulting

in the following theorem.

Theorem 3.1. (non-trivial) ii)

Theorem;

This proves that

i)

The Euler product

zeros in the critical strip As an entire function,

L(s,~)

has an infinite number of

0 j Re(s) j i;

L(s,~)

has a factorization

of the type

L(S,~) = eA(~)+B(~)S]T(I - ~)e s/p, P

where

A(w)

and

B(~)

are constants depending on the automorphic

representation

151

and the product runs over all the zeros of

iii)

L(s,~)

in the critical strip;

The sum

~i~i-i-n P

extended over all zeros of

L(s,~)

converges for all

q > 0

and diverges for

= 0.

Remark 3.1.1.

The location of the critical strip for

L(s,~)

immediate consequence of the non-vanishing of the Euler product for and the functional equation. therefore,

It will be shown in

using the functional equation at

L(O,~)

we also have

3.2.

= E(~)f(~)

§5

that

is an

Re(s) > 1

L(I,~) # 0

and

and let

N(T,w)

s = 0

½

L(I,~),

L(0,~) # 0.

The First yon Man$oldt Formula.

Let

~

be an automorphic representation of

denote the number of zeros of the Euler product 0 < o < 1

and

Itl ~ T.

GL2~A ~)

~(s,w)

in the rectangle:

For the sake of simplicity we assume that

coincide with the ordinate of a zero of

E(s,~).

T

does not

Recall that

r(s,~) = ~-½(S-Xl)r(i~2~1)~-½(s-X2)r(?),

where the infinity type infinite prime.

depends on the local representation

In particular the trivial zeros of

negative real axis when mentary series;

{11,12 }

~

~(s,~)

~

at the

are located on the

is a member of the discrete series or the comple-

this lost situation seems to never arise for

GL2~A ~)

and can

152

even be proved for those a u t o m o r p h i c r e p r e s e n t a t i o n s It

~

b e l o n g s to the principal series then

and the trivial zeros of

C(s,~)

%1

that have no ramification.

and

%2

are not purely real

are located on two rails parallel to the negative

real axis.

For any one of these cases w e have,

E-function,

that

from the n o n - v a n i s h i n g of the

2~N(T,7) = A R arg L(s,Z) + 0(i),

where

R

is the r e c t a n g l e in the

A R arg L(s,~) perimeter of term

0(i),

s-plane w i t h v e r t i c e s

is the v a r i a t i o n of the argument of R

in the counter clockwise sense.

w h i c h is independent of

7,

L(s,7)

3 1 ~ ± iT, - ~ ± iT, and as

s

traverses the

The n e c e s s i t y of adding an error

comes from the p o s s i b i l i t y of

having at m o s t a couple of trivial zeros inside

~(s,~)

R.

The c o n t r i b u t i o n of the left half of the contour is obtained from that of the right half by using the simple relation, w h i c h is a consequence of the functional equation,

arg L(q + it,q) = arg f(7) ½-s + arg L(I - ~ + it,~) + c,

where

c

is a constant d e p e n d i n g on the root number

8(7)

but not on

s.

We

then w r i t e

~N(T,~) = a L arg L(s,7) - ½ A L arg f(7) ½-s + 0(i),

where ! + iT 2

L

denotes the path going from 1 and then b a c k to ~ + iT.

1 ~ - iT

to

3 ~ - iT

Now w e clearly h a v e

A L arg f ( ~ ) ½ - s = -2T log f(~)

then

from

3 ~-

iT

to

153

and

A L arg ~-½(s-11).z-½(s-12)

also by the complex version

A e arg r(S-ll)r(s-12) 2 2

where

of

Stirling's

formula we have

= (2T - h Ti - 12 ) log ~T _ 2T - ~(!~ + % O1 + 12 ) + 0(T-I),

Ik = %k + iI , k = 1,2.

zN(T,x)

= -2T log z;

We then obtain

the formula

= (2T - h TI - 12 ) log ~T - 2T - ~ ( ½

+ h ~I + 12 )

+ 0(T -I) - 2T log ~ + T log f(~)

+ A L arg ~(s,~)

+ 0(i)

.Tf(~) ½. = 2T log < ~ ) - 2T + A L arg ~(s,Z)

where

+ 0(Ill

log T),

]I[ = max([%l],[%21).

To obtain the variation 2 auxiliary

of

arg ~(s,~)

along the path

L

we first prove

lemmas.

Lemma

3.2.

We have for large

T,

1 << log T, p 1+(T-y) 2

where strip.

the sum is taken over all the zeros

O = ~ + iy

of

~(s,~)

in the critical

154 Proof.

Logarithmic

differentiation

of the Hadamard

product

formula

F(s,Z)6(s,Z) = eA(z)+B(~)s~(I - ~)e s/P P

yields

~' - ~ (s,~)

r' --F'(s-12) - log z B(z) 1 i}. : ½ ~ (S-ll)2 + ½ F 2 - [{s---~p + p '

r v

in this formula we substitute large

Isl

in the angle

real parts of both sides

the bound

-~ + @ <

~

(s) = log s + 0(Isl-l),

arg s < z + 6,

some positive

is valid for

3 s o = ~ + iT

6,

-½ < o < ~3

and take the

and

t _> 2;

- P

this last inequality

evaluated

at

gives

[ Re{l+

P

1

So-P

where we have used the fact that

~] <

(s,z)

A'

log T,

is bounded

at

So.

Now by Theorem

we have

1 Re ~ = [ Re(0)Ip1-2 O O

= 0(i),

and also 3

[ ~e{ O

where

for

to get

-Re ~ (s,~) < A log t - [ Re{ _ P

which

valid

P = B + iy.

} = -

This proves

(~- ~)2+(r-y)2

the lemma.

--

p l+(r-y)

2'

3.1

15,5

The following

Corollary

is a useful corollary

3.3.

to the above lemma.

We have

i)

N(T + l,w) - N(T,w) < 1

ii)

A log T;

< A' log T.

IT- I~i I+(T-Y) 2 --

The next lemma, needed here to compute in the derivation of the explicit

Lemma 3.4. ~(s,Z)

and

For large positive

3 -½ < ~ < ~

~'

where

P = ~ + iV

Proof. 3 --+it 2

formulas

arg L ~(s,~),

in §4.

t

not equal to the ordinate

i (s,~) = [ s_--~ + 0(log t), P

We evaluate

the logarithmic

It-yl > 1

~I 1 p s-p

3.3

that satisfy

It-yl < i.

~(s,~)

of

at

one value from the other to get

p s-p

By Corollary

s = O + it,

derivative

~'(s,~) = 0(log t) + [{ i

the terms w i t h

of a zero of

we have

runs over the zeros of ~(s,~)

and subtract

will also be used

contribute

1

};

at most

I < ~

~3+ i t - p

1

3 ~ + it-p

i

0(log t).

p (t-y) 2

we know that the total contribution

p

I~+ it-pl

of the terms in

s

and at

156

with

It-yl < 1

is at most

It should

0(log t);

be observed

this then proves the lemma.

that in Lemma 3.4 the restriction

different from the ordinate of a zero

P = 6 + iy

that

t

be

is clearly not necessary

if

0#6.

To complete

the derivation

of the First yon Mangoldt

A L arg ~(s,~) = S Im ~

formula observe

that

(s,~)ds + 0(I),

(~)

where term

(~) 0(I)

where

3 ~ + iT

is a straight path going from denotes the variation of

~(s,~)

has no zeros.

arg ~(s,~)

to

~ + iT;

here the error

along the line

Re(s) =

3

Now

/ Im{--i }ds = A arg (s - p) (~) s-p

where

a

at most

denotes variation along the path 7;

proof of the following

Theorem 3.5. representation %1

and

and this is in absolute value

this remark together with the expression

Lemma 3.4 leads to the estimate

let

(6)

of %2

for

~

A L arg ~(s,~) = 0(log T).

(s,~)

given in

This completes

the

theorem.

(First von Mangoldt

GL2~A~)

of conductor

be the infinity

type of

Formula). f(~) ~

and and put

Let L(s,~)

~

be an automorphic its Euler product;

i%i = max(i%ll,i%21).

we have

N(T,z)

= ~2T-

log (Tf(z)½~ .~. - ~2T+

o(i~i log m),

where

N(T,~) = #{p = 6+i~f: L(p,~)

= 0, 0 < 6 < i, -T < t < T}.

Then

157

3.3.

Explicit Estimates.

The First von Mangoldt Formula given in Theorem (3.5)

depends implicitly

on constants whose values change with varying automorphic representations even if these have the same conductor and same infinity type.

It appears that these con-

stants can be estimated more explicitly if one imposes various restrictions on the local components

~

of the automorphic representation

P

7.

We want to consider

here the problem of making explicit the error term in the First von Mangoldt Formula for an automorphic representation whose local component at the infinite prime is a member of the holomorphic discrete series.

Let

A

and

k

be positive integers. let

F (A) o

~

defined modulo

A

with

subgroup.

~

be an automorphic representation of

Let

~(-i) = (-l)k;

Let

translates of a primitive holomorphic cusp form Recall that such an automorphic form

f(z)

be a Dirichlet character

be the Hecke congruence GL2~~ ) of type

generated by the {~,k,Fo(A)}.

f(z) has a q-expansion

f(z) =

oo ~ a(n)q n, n=l

with

a(1) = i;

also

f(z)

is an eigenfunction of the Hecke operator

T P

oo

oo

flTP = n~l a(pn)qn + ~(p)pk-i n=l~a(n)q pn, p~A,

and of the operator

U P oo flUp = n~la(pn)q n, Pl A,

with the corresponding eigenvalues being the of

~

is given by

a(p).

Recall that the Euler product

158

L(s,~)

= r(s,~)~(s,~)

with

F(s,~) = ~-½(S-%l)F(S-ll)z-½(s-k2)F(s~%----~2), 2 k-i ~I = -~--'

k+l 2 '

~2 = -

and

E(s,~) = p~A i "p~A 1 l-a (p) p-S i-~ (p) p-S+~ (p) p-2S'

where

~(p) = a(p)p½(l-k);

L(s,~)

satisfies

the functional equation

L(s,~) = g(~)A½-SL(l-s,~).

Recall that Deligne has proved

([ 7 ], §8.2)

~ ( p ) = kp + Op,

[%p[ =

that

[Op[ :

and Ogg ([25],p. 295)

has proved that when

plA,

can be defined modulo

A/p;

@

l~(p) l = p-½

if

p2~A

l~(p) l = 1

and if

@

if

1

for

~(p) = 0

can be defined modulo

equation and the Euler product imply, as we saw in L(s,~)

lies in

Theorem 3.6.

if

§2.4,

A/p.

f(z)

N(T,~)

and if A/p;

The line

L(s,~);

Re(s) = 1

the functional

that the critical strip

0 J Re(s) j I.

Let

~

be an automorphic representation

local component at the infinite prime belongs to the holomorphic let

p21A

cannot be defined modulo

is the boundary of the region of absolute convergence of

of

p~A,

be a primitive cusp form of type

be as in Theorem (3.5).

We then have

{~,k,Fo(N)}

of GL2~A Q)

whose

discrete series;

associated to

~.

Let

159

2T TA ½ 2T N(T,7) = -~- log (-~--) - T + (k41--) + 0(k) + 0(log (100Ak2T2)),

where the implied constants are absolute, meters that characterize

Theorem 3.7. of zeros

O

Js - ~] i 7

3

of

the automorphic

Let

L(s,~)

7

and

that is they do not depend on the pararepresentations

L(s,7)

7.

be as in Theorem 3.6;

counted with their proper multiplicity

is bounded by

4 -I (log 5) log (cAk2),

where

then the number

inside the circle

c = 2(27)

-2

3 4 5 2 ~(~) ~(~)

We give only a detailed proof of Theorem 3.7; the proof of Theorem 3.6 follows in outline the same argument used to derive the First von Mangoldt Formula except that at the crucial point one must replace the bound

A L arg ~(s,7) =0(logT)

by an explicit estimate of a type that will be given below.

Proof of Theorem 3.7. L(s,7)

inside the circle

formula to the circles

C1

CI: and

To get an upper bound for the number of zeros of 3 Js - ¼1 j ~

C2: Js - ¼[ ~ "

(~)mi

where

m

is the number of zeros of

maximum modulus of

L(s,7)

The estimates of

we apply the weak form of Jensen's we thus have

M

L(s,~)

inside the circle

Ogg for the

7,

a(p)

inside the circle

give

M

and

is the

C 2.

and of Deligne for the

and the expression

i ~(s,7)

C1

p~A (I-~(P) P-S) p~A (I-Xpp-s) (I-Opp-s)

%

P

and

o

P

160

5

1¢<¼,~)1

5

-

< ¢(5) 2.

Similarly,

from the identity

co

oo

~(S,~) = p~A(v~O(~(p)p-S)V)

we get for

s = O + it

and

co

p~A (v~O (%pp-s)v)(= w=0 ~ (Opp-S)W),

o > 1

]~(s,~)] < ~(o) 2.

To estimate the maximum modulus of apply the Phragm~n-Lindel~f by the lines

s = -½ + it

Theorem ([ 29 ], p. 195) and

s = 3 + it.

Mellin transform of a primitive cusp form

(3)

where type

~(s,~)

Since

inside the circle in the strip L(s,~)

f(z) of type

is the interval

~,k,Fo(A)}

[A-½,~],

{~,k,Fo(A)}

g(z)

s ÷ s + ½(k-l),

From the above integral representation S(-½,3)

it follows that

l~(s,~)l j c exp Itl c,

c.

We also have

bounded the

we have

3 2dy,

representation

~;

this is the formula in [37 ], p. 94.

and satisfies there

for some positive constant

we

is the primitive cusp form of

which is associated to the contragredient

modulo the change of variable

in the strip

and

S(-½,3)

is essentially

L(s,~) = f (y½kf(iy)yS + g(~)A½-Sy½kg(iy)yl-S)y (~)

(y)

C2

L(s,~)

is regular analytic

161

l~(3+it,~)l < ~(3) 2.

To get an estimate for

l~(-½+it,~)l

~(s,w)

we use the functional equation

E (~) A½-S (2~) 2s-l~ (l-s, ~)

F(l-s- k-l) F(s + k21)

thus

~(-½+it,~) I _< A(2~)-2~(~)2(t 2 +q-). k2

The PhragmSn-Lindelof Theorem strip

S(-½,3)

([ 29], p. 195) leads to the conclusion that in the

we have

k2 [~(s,~)[ j A(2~)-2~(~)4(t2 +-~-).

For

s

inside the circle

maximum modulus of

~(s,~)

C2: Is - ¼1 ~ 7

7

k2 t2 ~- + j 2k 2.

we have

inside the circle

C2

Hence the

is bounded by

M < A(2~)-2~(~)42k 2.

Finally we get

M 5

< cAk 2 '

with

3 4 ~(~) 52 c = 2(2~)-2~(~)

.

We now take the logarithm of both sides of Jensen's inequality to get

162

m < c' log (cAk2),

where

4 -i c' = (log ~) .

This completes the proof of Theorem 3.7.

Remark 3.3.1. inequality to ~(s,~) C2:

is -

gives

5

- iT 1

To complete the proof of Theorem 3.6 on

~ ~'7'

the

two

circles

the bound for

N(T + 1,7) - N(T,~) <<

el:

~(s,~)

log (cAk2T2),

5 Is - 7 -

with

s

one applies Jensen's

iT I < 13

-i7

and

in the strip

S(-½,3)

then

where now the implied constants are

absolute.

Remark 3.3.2.

The explicit estimates given in Theorems

(3.6)

were obtained under the assumption that the class one local components

and ~

(3.7) of

P the automorphic representation assumption

~

are all members of the principal series;

this

was in fact used in the form of bounds for the coefficients of the

Dirichlet series

~(s,~)

which in turn lead to explicit upper bounds in the

region of absolute convergence.

In order to extend the domain of validity of the

explicit estimates we are lead to introduce a new parameter which measures the global size of the automorphic representation.

This can actually be accomplished

by two seemingly different methods which we now proceed to sketch.

Method i.

Here we use the fact, established in

Ramanujan conjecture is true on the average: tation of

GL2~)

,

then there is a constant

if

~

§2.4, that the Petersson-

is an automorphic represen-

II~I] such that for all

x ~ 1

la(n) i << li~IIx, n<x

where the

a(n)'s are the coefficients of

independent of

~.

To get a numerical value for

to a real analytic cusp form line of the integral

~(s,~)

f(z),

and the implied constant is II~II, say when

~

is associated

one studies the growth behavior on the real

163

~o(N)If(~)l2(cI(O)y½(l+O)+ c2(O)y½(l-O))d~, where

N

is the conductor of

Cl(S)y½(l+s)

+ c2(s)y½(l-s)

~,

Do(N)

a fundamental

is the constant

domain for

Fo(N)

term of the Eisenstein

In the Fourier expansion about the cusp at infinity for the group already suggested in of

f(z).

§2.4

one can relate

II~ll to the Petersson

Once the average size of the coefficients

summation yields the estimate for

implicitely

of the local representation

Fo(N).

GL2~A~)

and

T ~ 2,

is known, partial

~

and the infinity type

Method 2.

and

Bounds for

3.6 and 3.7

on

~

equation of

we write the functional

has not been introduced erature of the subject.

-s

L(s,~):

L(I-s,~);

the dependence of the constants

clearly the global behavior of

where the

can be obtained from known estimates for

L(s,~) = g(~)f(~)

in order to make explicit

<< log (NII~[I~2T),

representation

% = max(l%li,l%2i).

~(s,~)

the F-function and the functional

{%1,%2}

This estimate can then be used to show, following

N(T+I,~) -N(T,~)

implied constant is absolute,

As

inner product

the argument in the proof of Theorem 3.7, that for any automorphic of

E(z,s)

It should be observed that this estimate

the conductor of ~ .

series

o > 1

where the implied constant is absolute. already contains

la(n) I

and

L(s,~).

that appear in Theorems

equation in a form that exhibits more

For this we use a bit of notation that

earlier in these notes but which is implicit in the litLet

~

be the subgroup of diagonal matrices

which appears in the Iwasaw decomposition;

if

o E SL20R )

we write

of

SL 2

164

o : (1 °
0

0

Let ~

be the Lie algebra of

A

e-OH(O)]k(@)"

and ~

its dual.

Let

~

be the nontrivial ^

element of the Weyl group of Let

A(N)

A

with

be the domain of

~(~) = -I

for any complex number

% C~.

defined by

A(N) : {(~ 0a_l): a2 _> N-½}.

Let

d*a

be a Haar measure on the group

(3)

used in the proof of Theorem 3.7

~.

Then the integral representation

can also be written in the following

alternative form.

Theorem 3.8. conductor tation.

f(~) If

Let

~

be an automorphic representation of

and root number

% E ~

g(~);

let

~

GL2~A ~) of

be its contragredient represen-

then we have

.l+l L(~--,~) = f (~(a)e %H(a) + s(~)f(~)½~(%)~(a)e~(%)H(a))d*a, A(f(~))

where

~

(resp.

~)

is a vector in the representation space of

chosen so that the corresponding primitive cusp form the isomorphism of

§1.3

~

(resp. ~)

f(z) (resp. g(z)) given by

has its first Fourier coefficient about the cusp at

infinity equal to one.

To recover the functional equation from this integral representation we observe that

2

is the identity element in the Weyl group and

The key point to observe here is that the functions defined behavior in the domain

A(f(~)).

~

and

~

g(~)g(~) = i. have a well

In fact, if we define

II~II = max( max If(iy)e2~YI, max Ig(iy)e2~YI) , yN~__> 1 y~__>l

165

then the above Theorem or the simpler

expression

<_

where

r(a,x)

(3)

give

+

is the incomplete

(2~)

_

},

f'-function

oo

F(a,x)

= / ta-le-tdt, x

and

k = ~weight "0

of

f

There are several well known assymtotic to explieitely automorphic a(z).

it.

representation

Here

therefore

estimate

k = 12

and

if if

f f

is holomorphic is real analytic.

formulas

for

F(a,x)

Let us now give an example. of

GL2~A ~)

N = 1

f(z) = g(z) = A(z).

corresponding

and also,

~

Let

that can be used ~

to Ramanujan's

be the modular

is its own contragredient

and

In this case we have

li~ll = max y>l

: max y~l

IA(iy)e2~Yl

i~(l n=l

- e-2~ny) i24

= 1

and so

L (.l+s __~_~)

To go much further functional

i _< { F ( ~ , 2 z ) ( 2 z ) - ~ ( 1 2 q < 7 )

+ ~.12-~ y t ~ , z ~ ).t.z.~. . )-½(12-~)~ ~.

than this one must look into the problem of approximate

equations.

form

166

We end this section by proving an inequality that is suggested by the methods of Stark-Odlysko which have already been quite successful in the problem of getting explicit estimates for the conductor of an Artin L-function.

Theorem 3.9. conductor

f(~);

let

f(s) = ~(s,~)~(s) 2

Let

~

L(s,~)

be an automorphic representation of be its Euler product.

satisfies for

Re(s) > I

GL2~A~)

of

Suppose that the function

f, -Re ~ (s) ~ O.

Then we have for

Re(s) > 1

Re { ie}

-_< 2R

{i}+_

½ l o g f(~) + Re {½ TF'(s-~-2 +~T" F''s-%2"(__~)+~'(½s+l)}.

P

where the sum

E P

runs over all zeros of

f(s) = 0

and

p

and

~

are grouped

together in the sum and are counted according to their multiplicities.

Proof.

(J-P. Serre).

From the Hadamard product formula

L(s,~) = eA(~)+B(~)S~T(I

- ~)e s/p

P

we have by logarithmic differentiation

(s,~) = ½ T

2

+ ½ "~ (

) - log ~ - B(~) - ~ p

and from the functional equation we have

nT

~(0,TT) = B(Tr) = - log f(TT) - B(~) _ ~~ { l ! P

+ l}p

or equivalently

B(~) + B(~) + log f(w) = - ~ [ P

+ %},

+ i} O

167

where

~ is extended over all zeros of L(s,~). Now we observe that if O a zero of L(s,~) then 1 - p is a zero of L(s,~) and similarly if p zero of

L(s,~)

then

~

is a zero of

L(s,~).

p

is

is a

Hence we can write

Re B(~) + ½ log f(~) = - ½ [@ {i@ + ~}i

and

(4) I R e { _ } - Re ~~' (s,z) = ½ log f(z) - log ~ + ½ Re ~F' ([~%1 ) + ½ R e ~ £' (s-~2), P where the

p

and

~

terms are to be grouped together.

of Riemann's zeta function ([ 6 ], p. 88)

1 s-p

Also from the theory

we have

%' 1 - - - ½ log ~ + ½ ~ ~ (s) = s-i

(~ + i),

P where the

O

and

~

terms are to be grouped together.

Adding to (4) twice the

real part of the last equality we get

[ ~e{ s_--i} - Re{ 2~' (s) + 7~' ( s , O } P

= 2 Re{sll } + ½ log f(~) - 2 log ~ + Re{F'(½s+l) + ½ T (

The positivity of the function

f~ - Re ~ (s)

) + ½-F (

) }"

now gives the desired inequality.

Although we have not used the assumption f, is essential~ in fact to verify that - Re ~ (s) ~ 0

Re(s) > i, in practice this one must employ Dirichlet

series whose domain of absolute convergence lies to the right of the line Re(s)

=

i.

Example.

(J.-P. Serre)

Let

~

ciated to a holomorphic cusp form of type

be an automorphic representation asso{~,ro(A),k}.

Recall that the infinity

168

type is 3 s = ~

and

%2 = -

0 < o < i.

b o u n d e d by

~

~ = ½

then

over the rationals,

if

k = 2

ml -< 1.7 + ½ log A. rkE

and If

E

L(s,~) m½

L(s,~)

in the

on the real line is

is the m u l t i p l i c i t y of a

is an elliptic curve defined

denotes the rank of the M o r d e l l - W e i l group of

E

then the above estimate together w i t h the Birch and Swinnerton-

Dyer c o n j e c t u r e suggests that E.

In the inequality of T h e o r e m 3.9 take

Then the number of zeros of

3 .24 + ~ log --(k2A);

over the rationals and

of

k+l 2 "

and let the sum on the left run over the real zeros of

interval

zero at

k-i 2

%1

rkE < 1.7 + ½ log A,

where

A

is the conductor

169

.§4.

Explicit Formulas.

4.1. The Second von Mangoldt Formula.

Our aim here is to prove a generalization, representations

of

GL2~)

,

in the framework of automorphic

of the famous Second von Mangoldt Formula of prime

number theory

[ p

where

~ P

log p = x - Z ~ x-L- - ~(i)(0) P p ~(0)

½ log (l-x-2),

<x

runs over the non-trivial

zeros of

((s).

We shall actually obtain a

formula which involves only a finite number of zeros and an explicit error term. As was already hinted in the introduction

such formulas

principle for automorphic

of other adele groups;

representations

can also be obtained in

we develop a general type of zeta distribution which generalizes formulas of interest

Throughout of conductor

f(~)

the Euler product

of the Dirichlet

all known explicit

in number theory.

this section

~

and infinity type L(s,~)

the 2-dimensional

in fact in §6

of

~

representation

is an automorphic {~i,%2}.

GL2(~).

of

GL2~A ~)

When convenient we shall write

in the equivalent of

representation

Let

form A(n)

L(s,~,r),

where

r

be the n-th coefficient

series

co -

observe

that

A(n)

--~'(s,~) :

[ A(n)n-S; n=l

has its support at the prime powers; define for

~(x,~) =

~

x > 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
-i

) + eT-11a(x) l,

n=l

where

1 c+iT ~'(s,~)~}ds, l(x,T) = 2--~ / {c-iT

and the term

cT-IIA(x)l

is to be included only if

x

To estimate the sum in the right hand side of contribution

arising from terms with

n _< 3x

and

is an integer.

(i) we consider the

n <_5x.

remains bounded and the resulting sum can be majorized by

T-I~ [ IA(n) ln-~. n=l

For these

flog xl-i

171

Recall that if the local representation of

is class one, then, in the notation P

§1.4, we have

det(l 2 - p-Sr(gp))-i =

~ a(pV)p -vs v=O

and

oo

--d l°gds det(12 - p - S r(gp))

=

~ A(pV)p-VS; v=l

by comparing power series expansions we easily obtain

A(pV) = ~a(p)log p -a(pV)log p - ~(p)a(pV-2)log p

where

~(p) = det r(gp).

if if

v = i v ~ 2,

If we assume, as we have done throughout these notes,

that the central character

~(p)

is unitary, then we obtain the following useful

inequality:

]^(pV) I

(2)

<

--

{la(p) llog p la(pV) llog p + la(pV-2) llog p

if if

v = i v ~ 2.

It can also be verified that this inequality holds even when the prime the conductor

p

divides

f(~).

From (2) we have oo

co

iA(n) in-C < ~[a(p) l(log p)p-C + ~ ~ la(pm-2) flog p)p-mC n=l

p

P m=2 co

<

co

[ la(n) l(log n)n -c + [ log p{p-2C + p-2C ~ la(n) i(log n)n c} n=l

p

< 3( ~ la(n) l(log n)n-C)(~(log p)p-2C). n=l p

n=l

lV2

Using partial summation and the estimates

fa(n) l

for

of

§3.3 we obtain

n<x co [ IA(n) In-c << llzll(log x) 2, n=l

where the implied constant

is absolute.

It remains to estimate

the contribution

to the sum in (i) arising from

the sum

S =

[ 3

li(n) Imin(l,T-lllog xl-l).

<

<5

7x_n_Tx

If we knew that the Petersson-Ramanujan tation

~

Davenport

was true then a relatively ([ ~ ], p. 113)

right hand side of

conjecture

< x >

would lead to the estimate,

constants are absolute. the Petersson-Ramanujan 3 ~x<

x + min(l, T--~-~x>},

the

inequality

x

to the nearest integer and where the implied

To obtain an estimate for

S

without

the assumption of

conjecture we are forced to take a different route.

n < x;

We first divide the sum satisfies

for the whole sum in the

(i),

is the distance from

treat the case

represen-

simple argument similar to that given by

<< II~II (log x){ x l ° g x T

where

for the automorphic

S

the

other

case

into two sums (a)

or

(b)

can be

Sa

and

below:

a)

T-i llog--Xni-i < N < I,

b)

T-iIlog ~I -I ~ ~,

treated

sb

by a similar

We

argument.

depending on whether

n

173

where

N

is a fixed number

to be selected later.

In case

n

satisfies

(a)

we

have

Sa

-n < x

If we replace

[A(n) l

by the inequality

in (2) and use the estimate of

§3.3 we

obtain

s a << N ll~IIx log x.

The sum

sb

runs over those

n < x

for which

!

> log (~) Tn -n

or equivalently

n _> x exp {- T!}.

For these that

n,

(~T) -I

the sum

sh

is bounded,

has at most

1 x(l - exp {- T~})

then the number of terms in

If we now apply the Cauchy-Schwarz

c(n)[A(n)[ 3 -~x
inequality

terms, and if we assume

sb

is at most

to

sb

1 << --'TN

we get

<< ( ~ c(n)2)½( ~ A(n)2) ½, n< x n<x ---

1

where A(n)

c(n)

is

i

if

by the inequality

x exp{- ~ } (2)

< n < x

and

0

and use the estimates

s b << (xN-iT-l)½(log

otherwise. of

§2.4

x) llxllx½.

and

Again we replace 3.3

to obtain

174

We thus have

S = S a + sb

<< II~llx(log x){n

We now select

N

so that

~ = N-½T -½,

~-½T-½}.

+

that is

1 n = T3

•

with this choice we have

i

s << 11~11x(1og x ) r - Z

A similar argument works for the sum in the range the above estimates and using the fact that

x < n

< 5 ~x.

Putting together

IA(x) l << II~]l x log x,

we now have

i l~(x,7) - I(T,x) I << 11711 x (log x) 2T -I + [1711 x (log x)T - ~

<<

where the implied constant

To evaluate satisfying

the

the condition

I1~11

x

1 (log x)2T -~,

is absolute.

integral

l(x,T)

that for any

u

we take a positive real number in the set

Z(~ ) = {%1- 2n, % 2 - 2n:

we have

i IU + Re(u) l _> ~.

Let

Lo

U

n • ~,

n ~ 0},

be the contour consisting of the segments

175

L : -U < O < c, t = T : - U < (7 < c, t = -T

where

c = i + (log x) -I

of the zeros of

L':

O = -U,

0 < t < T

L':

(7 = -U,

-T < t < 0

L : c

O = c,

-T < t < T,

and

T

is s u i t a b l y chosen so as to a v o i d h i t t i n g any

~(s,~).

Cauchy's r e s i d u e theorem yields

l

(3)

v

s

o

¢(2) (log x + ½ ¢(i)(0,7)

w h e r e the first term, to be i n c l u d e d only w h e n

_

p

xp

~.

v-

{ -i-,

s = 0

is a zero of

~(s,~),

arises from the obvious Laurent expansion

~'

the sum sum

~

1 ~(2)(0,~) (s,~) = -- + ½ + ~ A(n)sn; s ~(i)(0,~ ) n= I

E runs over the n o n t r i v i a l zeros of ~(s,~) P runs over those trivial zeros % of ~(s,~)

consider now the integral in (3) integral a l o n g t = T

L

Z(~ ) n L o

§3.3

T - i < T < T + i

is

<< i,

where

= [1~11 log (f(~)(T + I~ll + I~2[>>"

# 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

<<

II~II;

IY-tl ~ 1

with

w e have

2-0 I(s-p)(2+it-p)

<< £.

3 < -IY-tl 2 '

N(T+I,~)

Similarly

- N(T-I,~)

<< ~,

for those terms with

and these are in number at most

<< ~.

~ (s,O =

~,

i s-p

+

shows

Iy-tl < 1

and

-i
0(~),

P

where

the sum

constant

Z runs over those p for which It-YI < i, P is absolute. From this representation we obtain

that

In conclusion we

(not coinciding with the ordinate of a zero)

~'

(4)

-P

to get

for the logarithmic

by

1 I 2+it-o

at most

further w e

t = T.

IF (2+it,~) I + ~{slo - - 2+it-p o

using the fact that

12 + i t - p] > 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 <<

~

,% + 0(%)

I~ l
uniformly for

-½ < ~ < c,

where the implied constants are absolute.

the contribution to the integral in

(3)

Therefore

along this part of the line

L

is at

most

c

<<

s

~2 f ]Xldc~ -½

2 -c << % T -lj x°do -½

<< ~2T-Ix.

A similar argument can be given for the corresponding part of the line estimate the contribution along the remainder of

l~'<s,~)l << ll~IIlog which is valid for any

s = o+it

with

L

To

we make use of

(f(~)(21s I +

l~ll + Im21)>,

O __< -½

]s+v] _> !4,

and

[.

for

v E Z(~ );

to prove this one again uses

~(s,v) = O(log(21sl +

l~ll+ 1121)) + ]~'(2+it,~)l+Z{~p

2+1_0],

P

the whole sum similarly for

Z is now estimated as before using the fact that Re(s) J -½; P ~ (2+it,~). The term log(21s I + 1%11 + I%21) results from

estimating the logarithmic derivative of for which

Is+vl ~

i,

F(s-%I)F(s-%2) 2 2

at points

s = o+it

v ~ Z(~ ).

The contribution of the integral along the remainder of

L

and

L

is

178

not larger than

-½

<<

T-l/

-U

<< £x

-½ -i T .

We therefore get

v

s

1 / {-$ (s,~) x-- }ds << xg~2T-I. 2~i L+L % s

The integral along the path

L' + L'

<< u-lll~llXog
is at most

lhl +

T -U

Ix21))/ x

dt

-T

<< TU-Ix-UII~IIIog(f(~)(2U +

If we now assume, as we may do, that as

U + ~.

I%1] +

I%21)).

x > I, then the last expression goes to zero

Putting together the above estimates we get for

P(x,~) = -(log x + ½ ~(2)(0'~)) ~ x0 C(1)(O, ~) IYT!T P

x > 1

1 x__l + R(x,T),

e z(~) x__

where 1 R(x,T) << x(ll~lllog{xf(~)(T + I%ii + I%21)})2T 3,

Z(~ )x = Z(~ ) - {0},

and the implied constants are absolute.

Since the product

179

p(S-Xl)p(s-X2)~(s,w) 2

is entire and that

~(s,~)

Re(>`I) ~ 1

2

is free of zeros to the right of Re(s) = 1

and

Re(% 2) ~ i.

discrete series we have

Re(h i ) J 0.

In fact, if If

~

~

it follows

belongs to the Holomorphic

is a principal series representation

then Rankin's trick applied to the Euler product

L(s,~)

as in

§2.4

using the positivity of the coefficients of the Dirichlet series Re(>`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_ >, << 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) << ~(ll~rlllog {xf(~)(m

I>11

+

+

IX21)))2T ~-,

and

o~

xkl_2n

( oo) : n:O Oh--7

xk2-2n

+

<< xl(O){log(l+x -2) +max(Ill{ + IliI-l, II2 ] + ix21-1)},

I(o) = max(Re(Xl) , Re(X2));

furthermore all the implied constants are absolute.

(2) Remark 4.1.1. be included only if the terms with

4.2.

As already observed, the term

[(s,~)

~i = 2n

and

has a zero at s = 0; X2 = 2m

(log x + ½ " [ (0,7)) ~(i)(0,~)

the sum

S(~ )

is to

does not contain

if they exist.

Examples of Explicit Formulas.

If

~

is an automorphic representation of

2-dimensional complex representation of

GL2(¢),

GL2~),r

X

is the

its character and

[(s,~) = ~ d e t ( l 2 - p-Sr(g_))-i P P

is the associated zeta function, then we have for a prime power

A(p n) = x(g~)log P.

If we now let

x

be fixed and let

T ÷ oo in the formula for

~(x,Z),

obtain the explicit formula

[(2)(O'~r)) - ~ -x p + S(~ ), T(x,~) = - (log x + ½ ~(i)(0,~)

p

then we

181

where the sum

E

is to be understood as

P

Now let p~A

E

lim T+~

[

xP/p.

1Y[ j r

be an elliptic curve defined over

~

of conductor

A.

For

let

Card EOFp) = p + I - a(p),

where

a(p) = Tr(~p)

is the trace of Frobenius acting on

count the number of points of

E

%-adic cohomology,

defined over the p-element field

~

.

Suppose

P that the Hasse-Weil zeta function

L(s,B) = (2~)-SF(s)~(s,E)

=

(2~)-SF(s) ffA ( I - a ( p ) p - S ) - I v (l-a(p)p-S + pl-2s)-i p~A

is actually the Euler product conjecture ([ 40 ], p. 156)

L(s+%~2,~) of an automorphic representation as Weil~

suggests.

Suppose also that the Birch and Swinnerton-

Dyer conjecture is true, that is to say, the multiplicity of the zero of at

s = 1

is the rank of the Mordell-Weil group

E(Q).

L(s,E)

Then, under these restric-

tive assumptions, we can specialize the explicit formula of Theorem 4.1 to yield

Tr(~)v log p = -x rank E(Q) - [

p#l

p~< x

where the sum

E P

critical strip

runs over the non-trivial zeros

f(~)

~(i) (O,E)

p

of

~(s,E)

inside the

3 ½ < Re(s) < ~.

Finally if conductors

xO ~(2) (O,E) ~ - ½" - log (x-l),

and

~

and f(~')

~'

are automorphic representations of

and if

GL2~A ~)

of

182

L(s,~ x z ' , r @ r )

4 = ~ _ ½ ( s _ % i ) F ( ~ ) ~4( s , ~ _ z i=l

is the hybrid Euler product associated to them in

x ~ ' ,r@r)

§2.3, then one proves by a more

elaborate analysis an explicit formula of the same type as in Theorem 4.2.

More

precisely, if

(s,~Xz',r®r)

x(g~)x(g~n)(log p)p-nS,

= P

then, for

x > i

and

n

T > i, we have

xp Z X(gp)X(gpn)l°g P = x6(~,~')- Z m ( p ) ' ~ + S(~ , "iT') + R(x,T), pn<_x P

where the sum ~(s,~X~',rOr)

E runs over all the nontrivial zeros (and poles) p of P with IImpl j T; each counted with its proper multiplicity

4

m(p);

oo x%i_2n

S(~°°'~') = i:l~ n~O= Xi-2n '

and

_i

~(x,T) << ll~lloxT](log{xf(~)0(T

117110 max(ll~ll ' II~II)2'

f(~)O = max(f(~),f(~'))

=

Here

6(~,~') = i

if

~'

+

and

is the contragredient of

~

%0)})2,

%0 =

l~ll

+

Ix21 I~31 I~41" +

and zero otherwise.

+

The

key point in the derivation is the explicit estimate

N(T +

1 , ~ x ~ ' ) - N ( T , ~ x ~ ' ) << ll~llolog(f(~)o(T + %0) )

for the number of zeros (and poles) critical strip with

T < y < T + i.

p = ~ + iy

of

L(s,~X~',r@r)

inside the

183

§5.

5.1.

Zero Free Regions for

A Hadamard-Landan

L(s,w).

Type Inequality.

Our purpose in this section is to derive zero free regions inside the critical strip

0 ~ Re(s) j 1

for the Euler products

L(s,~).

The possibility

of obtaining such zero free regions was suggested by Rankin's work

[ 30 ] where

it is shown that the Euler product

L(s-y,~)11

= (2~)_SF(s)~(l-T(p)p

-s+pll-2s)-l,

P

associated with Ramanujan's

function

%(n),

13 Re(s) =--~

does not vanish on the line

An attentive reading of Rankin's article also suggests how his method, with a few modifications,

may give the non-vanishing

to any automorphic

representation

of

of the Euler products

GL2~)

on the line

result that we prove below which is comparable

the only

L(I,z) # 0,

simple nature.

L(s,~)

for Dirichlet L-functions.

convinced that the methods also work for adele groups other than convolution method is available,

but

The main

(see Theorem 5.1) gives a zero free region for

to those that are possible

an analogue of Rankin's

associated

Re(s) = i;

point that may not be clear in Rankin's method is the proof of this only requires an extra argument of a relatively

L(s, )

We are

GL2~),

in particular

where

for

GLn~)-

Throughout tation of

GL2~ ~ )

Euler product

this section we assume that

~

of conductor

type

L(s,~).

given in Rankin

We now proceed

~

where

~

represen-

and associated similar to that

of the logarithmic

We will treat only the case of an automorphic

which is formally real,

L(s,~) = L(s,~),

{%1,%2}

to derive an inequality,

[ 30 ], which will give the positivity

of certain Euler products. sentation

f(~), infinity

is an automorphic

that is to say a representation

is the contragredient

of

~.

~This has now been proved by Jacquet and Shalika in [ 15 ].

derivative repre-

satisfying

184

W i t h o u t loss of g e n e r a l i t y w e may assume that all the local components of

~

are class one representations,

of

~

has the form

and in p a r t i c u l a r that the zeta f u n c t i o n

~(s,~) = ~ ( l - a ( p ) p -s + @(p)p-2S)-i P

= ~(l_%(p)p-S)-l(l-~(p)p-S) -I. P

This is p o s s i b l e b e c a u s e

~

contains at most a finite number of r e p r e s e n t a t i o n s

that are not class one and the local factors a s s o c i a t e d to any one of these can have at w o r s t only poles.

Let

%(p) = exp i (p), n(P) = exp -i@'(p),

where

8(p) = ~(p) + iB(p)

real numbers.

~(s,~®~)

and

@'(p) = ~'(p) + iB(p)

and

~,e'

and

B

are

W e also need to consider the f u n c t i o n

= ~(2s)~(s,~X~)

=~(i-

% ( p ) ~ ( p ) p - S ) - l ( l - % ( p ) ~ ( p ) p - S ) - l ( l - ~(p)n(p)pS)-l(l-~(p)~(p)pS) -I.

P

F r o m the results of Re(s) > ½ ~(s,n®~).

except at

§2

w e know that

s = i

~(s,~ × ~ )

w h e r e it has a simple pole;

W e use these functions

to show that

enough to show this for the m o d i f i e d zeta ~(i,~) = 0.

s = i

the same holds

L(I,z) # 0.

~(s,~).

Now Rankin's c o n v o l u t i o n method

has a double pole at

is a n a l y t i c to the right of

Suppose

(§2.)

true for

Clearly it is ~(i,~) = 0, then also

shows that

~(s)~(s,~®~)

and h e n c e the f u n c t i o n

H(s,~) = ~ ( s ) ~ ( s , ~ @ ~ ) ~ ( s , ~ ) ~ ( s , ~ )

is r e g u l a r in a n e i g h b o r h o o d of

s = i.

O b s e r v e that

~(s,~)

can h a v e at w o r s t

185

a simple zero at s = i;

s = 1

for otherwise

but this contradicts

the function

the fact that for

H(s,z)

Re(s)

w o u l d have a zero at

> 1

co

H(S,~)

=

~ A(n)n -s, n=l

A(n) >__ 0

as can easily be seen from

-hE

log H(s,~)

[

=

[ ]i+~
p~f(z)

convergence

coefficients

imply that the real point on the line of absolute

ticular

Clearly

H(I,~)

s = i.

# 0

H(s,~)

H(C,~)

# 0

for

and log H(s)

Recall Landau's

Theorem:

for

n

The absolute

singularity.

of

n= 1

Re(s)

> 1

and the positivity

o > abscissa

is regular

convergence

of convergence

is a

and in par-

in a small n e i g h b o r h o o d

if the Dirichlet

of its

of the point

series

co

F(s) =

where

c(n) ~ O,

gO < c ~ c I

and

F(s)

~ c(n)n -s n=l

is analytic

of the real axis,

for

C > C1

in a n e i g h b o r h o o d

of the segment

then

co

F(s) =

Now let

g0

Landau's

Theorem,

~ c(n)n -s n=l

be the first real zero of the Dirichlet

for

H(s,~),

C > 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(~) < --

<<

[ p]f(~)

(io$ p)[a(p)Ip -O l-]a(p)]p

--O

log f(~).

Similarly we have

{'

(3)

-~ (s,{)

For

{(s),

1

= -~ s - ~ + O(Z). P

modified at the local factors corresponding

to primes

plf(z),

we use the well known result

(4)

_ !'

1 [ i + O(log ~ (s) = s---']-s-----p P

f(~)(l

We did not derive in §3 a result for there can be used without much difficulty

(5)

-

We thus obtain,

~'

(s,~)

1 s-i

+ I~))

"

6(2s)~(s,ZX{)

to prove

1

[ --

P s-p

+ o(~).

on taking the real parts in (2),

(3), (4) and (5), for

Reo > 1

(6)

but the arguments

-Re ~H' (s,~) <__2 R e { l } _ ~

R e { l } + ci%, O

188

where

the sum

l p

runs over the zeros

IY - tl < I; observe zeros as we want. positive of

that since

sponding

Re(s-p)

> 0,

Here and in the following

and absolute.

E(s,~).

P = B + iy

We now choose

By retaining

to the zero

t

of

we may retain the constants

of

L(s,~)

and

(6)

L(s,~)

with

3 ReD > ~

and

in the sum as few Cl,C2,...,etc.

to be the ordinate

in the right hand side of

B + iy

H(s,~)

¥

of a zero

only the terms

we obtain

are B + iy

corre-

in the region

t _> c0~(1) -I

(7)

-Re ~

We omit all zeros and poles and

in

(o + it,n) _< -

H' (o + 2it,~)

+ c2Z.

in

(6)

and obtain

for

t _> c0%(i~i

~ > 1

H t

(8)

Also from

-Re ~

(o + 2it,~) ~ c3~.

-Re ~

(0,7) <_

(6) we have

(9)

The three estimates inequality

(7),

(8) and

(9)

when substituted

(i) give

6 0 _< o-i

Take

+ c4~.

o = 1 + 6~ -I,

where

6

8 o-B + c5~"

is a positive

< 1 -

( 86

constant.

Then

_ 6)%-i

6+6c 5

and if

6

is suitably

chosen with respect

to

c6 ~3 < 1 - ~ - ,

c5

we have

into the basic

189

where

c6

is an absolute

For conjugate then

0

<

t

small but different

from zero we must exploit

the presence

zeros above and b e l o w the real axis in a small n e i g h b o r h o o d t < c0%(i)-i

also a zero of ~(s,~).

constant.

and let

~(s,~);

similarly

Again w e have for

- Re ~

(10)

p = $ + iv, y = t, ~ = $ - iy

of

i.

be a zero of ~(s,~);

is a zero of

~(s,~)

of

it is

and of

o > i

(O,~) - c7~ < 2 - 2 Re{ -- o-1

} - 2 Re{

}

d-1

2 < -- o-I

provided

the following

condition

2 o-$

is satisfied:

(*)

t < o - B.

Similarly we have

(Ii)

- Re ~

(o+it,z)

- c8% <_ 2 R e { ~ } - 2

Re{

}-2

2

2

2(o-$)

d-1

o-6

(o_~)2+4t2

2 < • -- o-1

_

Re{

1

121 5 o-6

- - , - -

H v

(12)

- Re ~

(o+2it,~)

- c9% i 2 R e { ~ i}

- 2 R e { ~i }

< 2 -- o-i

2(o-B) (o_B)2+t2

2 < --o-i

61 5o-6"

2(o-B) (o_B)2+9t2

Let

- 2 R e { ~ i}

190

The estimates

(i0),

(ii) and

(12)

and the basic inequality

(i) give

16.8

~--B ! ~

If w e take

o = 1 + ~-i,

+ Cl0 ~"

we get

< i - 2i--7'

where O

6

-i

is chosen p o s i t i v e and

<_ .32

Now o b s e r v e that for the chosen

Cl0

we have

o - B _> ~ ( i . 0 4 ) .

Hence the condition

(*)

6

is a u t o m a t i c a l l y s a t i s f i e d if

It remains to c o n s i d e r the s i t u a t i o n w h e n

t < --

t = 0.

H e r e w e can no longer

use conjugate zeros but w e can appeal to the earlier a r g u m e n t using only the function

H(s,Z).

zero.

Let

~

We w a n t to show that near and

~

fundamental inequality

be two zeros of

s = 1

there can b e at m o s t a real simple

H(s,Z)

near

s = i;

then w e h a v e by the

(i) and (6)

H ~

0 i - Re~ (o,~) 2

2

i o-i

o-~

2

o-~ + Cli ~ l j~"

or e q u i v a l e n t l y

o-~ +

If

O

is taken to be

i + 60%(i)-i ,

60

this last i n e q u a l i t y shows that

<--

+ c12%(i) "

for a s u f f i c i e n t l y small p o s i t i v e constant B

and

~

cannot b o t h be greater than

191

1 - 6'%(i) -I regions for

for a suitable positive L(s,%)

when

% # ~

6'.

The problem of getting zero free

is somewhat simpler and follows more closely

the classical derivation of a similar result for Dirichlet L-functions with

X

L(s, X)

a complex valued character.

We collect our results in the following theorem.

Theorem 5.1.

There is an effectively computable absolute constant

with the following property. of conductor

f(~),

the Euler product

If

infinity type L(s,~)

~

is an automorphic representation of {11,12 }

and norm

l[zll and if

c13

GL2~~ )

z # ~,

then

has no zero in the region defined by

O ~ 1 - c13%(t) -I

if

Itl ~ 1

~ i - c13%(i)-i

if

Itl ! i,

and

where

%(t) = ll~H2(log f(~)(Itl + I%11 + I%21)).

If

~ = ~,

the only possible zero of

L(s,~)

in this region is a single

simple

zero. Remark 5.1.1.

If we assume the Petersson-Ramanujan conjecture for the

automorphic representation

~

then we can use the somewhat simpler inequality

12 Re ~ (O+it,~)l i -2 ~ (O) - Re ~ (O+2it) - ½ R e ~ ( O , ~ x ~ >

which is obtained from the trigonometric identity

192

14

Remark 5.1.2. contragredient, products

cos A cos

If

~

BI

< 2 + cos 2A + cos 2B.

is an automorphic

then the above estimates

L(s,~X~)

representation

and

lead to zero free regions

similar to those obtained in Theorem 5.1.

~

its

for the Euler

For this one needs

to consider only the function

g(2s)~(s,g xZ)

Z lZ(p) n

= exp{ p#f(z)

If

~

and

7'

are

one can show that

two arbitrary automorphic L(l+it,~xT')

# 0

+ q(p)nl2 P -sn/n}-

n=l

representations

for any real

t.

of

and uses the information about L(s,~ x3)

5.2.

x~')L(s,~' x~')

just mentioned.

Prime Number Theorems.

If

~

is an automorphic

representation

of

GL2~ ~ )

co

~(S,g)

-

=

[ A(n)n -s, n=l

then the explicit

formula of

T(x,O

§4

gives,

=

for

x > 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) <<

xr-~(ll=iI log

(xf(~)(T+

I~11+

I~21))) 2

and

Is(~)l

<< x % ( ° ) { l o g

(l+x-2) + max (l~ll

+

IxlJ-l, lx21 + l~2 [-1)}"

The zero free region given in Theorem 5.1 now yields

[xP[ << x exp{- cl3£(T)l°gx}

where

£(T) = ll~ll2(log(f(~)(T+l~iI + l~21)));

by the estimate of

§3

and Theorem 5.1

i

*

i << £(T) 2

IY
where we have excluded the exceptional zero companion

i-8.

if it exists as well as its

From this there results the estimate

x6 (12)

B

~(x,~) + ~ - +

xl-~_l

i-6

- s(~) Cl31og £(T)

x}

as a function of

x

<< x(ll~ll21°g(xf(w)(T+ 1111 + 112 l)))2(exp{-

To select a suitable value for

T

impose some condition on the size of the conductor

f(w)

we suppose that

h f(~) < exp{cll~ll-l(log" " x) ~},

-1

" + T 5).

we must first

in relation to

x.

If

194

where

c

is any positive constant and choose

T + Ixll + IX21 : exp{- cll~ll-l(log x)½},

then all the terms on the right hand side of

are of the order of magnitude

<< xll~H2exp{-c, ll~II-l
(i3)

where z.

(12)

c'

is a constant depending on

c

but not on the automorphic representation

We also have

xl-S_l i_~ = x

for some

o

between

0

and

I-B

log x

and the last expression is less than the

expression in the right hand side of

(13).

We have thus proved the following

result.

Theorem 5.2. conductor

f(~),

Let

~

be an automorphic representation

infinity type

{Ii,~2}

and norm

II~ll. Let

of c

GL2~ % )

of

be any positive

constant and suppose

f(z) ! exp{clIzll-l(l°g x)½}"

Then

6 A(n) = _ ~

+ 0(xl(O)max{Ill I + IIi I-l,ll21 + 1121-1})

n<x

+ O(xll~II2exp{-c, ll~ll-l(log x)½}),

where c

I(o) = max(Re(ll),Re(12));

c'

is a positive constant depending only on

and all other implied constants are absolute.

Here

6

is the real zero of the

195

Euler product

L(s,~),

if it exists, satisfying

6 >l-6(ll~l121og(f(~)(l+llll+IX21))) -I.

The elementary estimate

A(n) = ~ X(gp) log p + n<__x p <_x

([l~ll~=(logx)2),

which is deduced from the inequality given in §4.1 leads to the following result.

Theorem 5.3.

Assumptions and notation as in Theorem 5.2.

If

- ~(s,~) = ~ X(gp)(l°g p)p-nS, pn

then

x B + 0(x%(O)max{[%ll + II1 l-1 ,1%2] + I%21-1})

X(gp) log p = - ~ pjx

+ 0([l~II2x exp{-c'll~ll-l(log x)½}).

Remark 5.2.1.

~A

If

A(2),

the Ramanujan modular form ii max(Re(~l),Re(~2) ) = - ~-

is the automorphic representation associated to then the conductor is

1

and

%(o) =

and hence Theorem 5.3 gives

ii p 2 T(p)log p << x exp{-c' (log x)½}. p<_x Remark 5.2.2. let

Let

f(~)0 = max(f(~),f(~'))

max(%(o),%'(o)). following estimate

~

and and

~'

be automorphic representations of

I1~1[ 0 = max(ll~ll , liT,I])

GL2~);

and let ~0 =

The same argument leading to the proof of Theorem 5.2

gives the

1 g6

(14)

X(gp)X(g~)~

log p = x~(~,~')

+ 0(x%0-% 0) + O(x exp{-c'(log

x)½})

pix

where

6(~,~')

~en

= 1

if

~ = ~'

Ramanujan modular

7'

is the contragredient

is the automorphic

to a-results

~

and zero otherwise.

representation

form, Rankin ([ 31 ], p. 247)

version of Theorem 5.3 without

of

associated

to the

had obtained a somewhat weaker

the error term and gave an interesting

for the size of the eigenvalues

of Hecke operators.

application

Other applica-

tions of the hybrid formula will be given in future publications.

5.3.

The Problem of Exceptional

Zeros.

Our program of eventually automorphic tation

representation

itself necessitates

utilizing

to obtain information

about the automophic

that we have explicit

information

regions inside the critical strip for tion about the possible exceptional an exceptional

the explicit formulas associated

L(s,~)

zero.

well known in number theory.

represen-

about the zero free

and in particular

further informa-

The proof of the non-existence

zero will have many far reaching consequences

to an

of such

some of which are

At present there seems to be three approaches

to

these problems:

(I)

To use the Saito-Shintani

theory of liftings [22]

tations to obtain an analogue of the Brauer-Siegal

(II)

To exploit more closely the implications

to non-abelian

class field theory.

morphic representations representations

of the Weil group

WL

and n-dimensional

effective Brauer-Siegel

estimates.

semi-simple

that the problem of exceptional

zeros arises only for a very special subclass of representations. tion one finds a natural interpretation

theory [ 16 ]

the relation between auto-

GLn~ ~ )

suggests

represen-

Theorem.

of Jacquet-Langlands

More precisely,

of the adele group

of automorphic

for some of the results of

For example if

f

In this direcStark [ 39 ] on

is a cubic polynomial

and

197

its splitting field and if

~K(S)

quotient tation of

K

over

is the Dedekind

~K(S)/~(s) GL2~A~).

~

has Galois group the diahedral

zeta function of a cubic subfield of

is the zeta function Now

~(s,~)

is a quadratic extension of

~(s,~)

Q,

computable

representations

associated

K,

of an automorphic

has an effectively

because it really comes from an automorphic L

group of order

on

G

then the

represen-

zero free region GLI~AL),

to a grossencharacter

where

of order

3.

In this situation one can apply the classical arguments of Hadamard to get effective estimates.

(III)

A far more interesting

consists in considering

suggestion has been given by Serre.

the Euler product

~(s)6~(s,~,r)7~(s,~,r2)4~(s,~,r3

where

r

square and

is the 2 dimensional r3

its symmetric

for the exceptional

representation cube.

~ of

of

GL2(~),

zero can be obtained if the zeta

GL2~)

),

r2

its symmetric

The claim is that an effective

approach is not entirely unrelated to representation

His idea

to other

(I),

~(s,~,r 3)

but involves

GLn'S.

upper bound

is entire.

This

lifting the automorphic

198

§6.

Zeta Distributions.

6.1.

Introduction.

In this section we let defined over

~

and let

~

G

be a connected reductive algebraic group

be a representation of the adele group

occurs in the space of cusp forms and let structed according to Langlands'

L(s,~,r)

recipe (§1.4).

the following fundamental question:

§6.2

be an Euler product con-

How is the functional equation o f

Weyl symmetry.

L(s,~,r)

associated to the data

we construct certain distributions depending on

Theorem (6.2)

which

The problem we study here concerns

reflected in the structure of the 'explicit formula' In

~A

~

and

r

{~,r}?

and in

we prove that these distributions possess a certain element of These distributions had already been considered by Weil [ 42 ] for

a certain class of Euler products that had a definite number theoretic interest. The idea of splitting the distribution obtained by Weil into two parts so as to reflect both sides of the functional equation does not seem to have been considered before~ our methods are nevertheless the same as those of Weil.

In a final section

we state a simple lemma that relates the location of the zeros (and poles) of L(s,~,r)

to the positivity of the distributions that appear in the explicit for-

mulas.

Due to our incomplete knowledge about ramification phenomena present in automorphic representations we are forced to introduce the following

Working Hypothesis:

~ = ~

fs an automorphic representation of the adele group P

~A; p

~

has a conductor set

f(~) = {p}

which consists of a finite set of primes

for which the local components are not class one representations;

complex representation of the dual group character; class in

for each prime cG C

p

associated to

not in ~

f(~)

CG~ let

as in §1.4;

of dimension {gp}

d

and

r

is a

X

is

its

be the semisimple conjugacy

for each prime

p

in the group

ci G~

in

f(~)

we

P assume the existence of a conjugacy class {g~}

which is the

199

dual group of a subgroup r'

of

CG~

associate,

G'

of dimension

of

G

d' ~ d;

and of a finite dimensional to the local component

~

representation and

r

we

as in §1.4, a F-factor

d

r(s,~oo, r ) = ~ - 2 ( s - X i ) r ( ~ ) ; i=l we let

L(s,~,r)

= £(s,~ ,r) p e ~f(~) det(id,-p-Sr'(g~))-i p ~ f ( ~ ) det(Id - P - Sr(gp))-i

be the Euler product associated

L(s,~,r)

with the understanding suitably modified. have a meromorphic functional

f(~,r)

that the local factors at primes

a number field

K

~n

f(~)

The key assumption about these Euler products continuation

and

have to be

is that they

to the whole s plane and in fact satisfy the

= g(z,r) f (~,r)

s(~,r)

the functional

L(s,~,r)

only over

p

-s

L(l-s,z,r) ,

is a rational number all of whose prime divisors belong to the f(~)

dK

(i d - p-Sr(gp))-i

equation

conductor set

where

{~,r}; for simplicity we write it as

= F(s,z , r ) ~ d e t P

L(s,~,r)

where

to the data

is a complex number of absolute value

Assume

Over

equation probably has to be taken in the form

I~_ S

= £K(~,r)(d d i m r f(~,r)) "2 L(l-s,~,r),

is the absolute value of the discriminant ~.

I.

L(s,~,r)

of

K/~.

Here we will work

is the ratio of two entire functions

at most a finite number of poles and is bounded in every vertical

each having

strip of finite

200

width.

Finally we assume that the coefficients

~(s,~,r) = ~ d e t ( l

in the expansion

d-p-sr(gp))-I

P co

=

satisfy for large positive

Z a(n)n -s n=l

x

Z a(n) << x. n<x

The only justification Hypothesis

that we can offer for introducing

is that all known Euler products

algebraic geometry seem to satisfy it.

that arise in number theory and

In particular

§2.3

are of this kind and therefore

6.2.

The Explicit Formula and Weyl Symmetry.

We consider real line

(A)

if

~

£

all the examples given in

the contents of Theorem

(6.2) are not empty.

of complex valued functions defined on the

that satisfy the following properties:

h(x) @ ~ ,

there is a real number

h(x)exp{(½+a')Ix I}

(B)

the class

the Working

h(x) C ~

is integrable

a' > 0

such that the function

on the real line;

and its derivative are continuous on the real line except at a

finite number of points

{~.}

where

h(x)

and

~(x)

have discontinuities

of the

1

first kind with

(C)

h(~i) = ½{h(~i+)

there exists a real number

h~x)exp{(½+b)Ix]}

= o
The properties

as

+ h(~i-)};

b > 0

Ixl ÷

such that

and

~.

that characterize

use freely, as we will do in

h(x)exp{(½+b)Ix I} = o(i)

the class of functions

the following without

stopping

~

allows us to

to justify it, the

201

inversion formula for the Fourier transformation; Lang's presentation of distributions

of

Well's result

we will also use, following

([ 19 ], chap. X),

some of the properties

in the sense of Laurent Schwartz [ 34 ].

We now define the Mellin transform of a function

h E ~

by means of

the integral

R(s)

we will consider

h(s)

i

(s-½)tdt;

as a function of the complex variable

priately we could consider algebra of rank

= /h(t)e

h(s)

s;

more appro-

as a function on the dual of a complex Lie

which is connected with the dual group

CG.

In order to

simplify our formulas we also consider the subclass of functions defined by

~o

= {he

: £(0)

= t{(1)

We now assume the Working Hypothesis consider,

for positive real numbers

a

sists of the boundary of the rectangle the line

Re(s) = ½

-T ~ Im(s) ~ T. ~0 = -a - iT f (a) If

for the Euler product T,

the contour

We let

a = l+a+iT,

~ = l+a-iT

be the corners of this rectangle;

sO = -a+

line joining

is not the imaginary part of a zero of the Euler product

Cauchy's residue theorem applied to the function

iT

and

and

~

we let

~

and

L(s,v,r),

h(S)d~ log L(s,~,r)

~. then

gives


L' f h(s) ~ (s,~,r)ds -

(~)

located about

-a < Re(s) < i + a

and

and

which con-

for a complex number

taken along the straight

/ h(s) ~L' (s,~,r)ds = (2~i) C(a,T)

L(s,z,r)

C(a,T)

in the s-plane symmetrically

and is defined by the inequalities

denote the line integral T

and

: 0}.

f

L' h(s) ~ (s,~,r)ds + o(i),

(~o)

202

where the sum

~

runs over the zeros (and poles)

p = B + iy

of

L(s,w,r),

Y counted with their proper multiplicities and whose imaginary parts satisfy IYl ~ T; the error term

o(i),

which results by integrating along the horizontal

rails and using the fact that the number of zeros with

T < Imp < T + 1

tion

of

where

T

a'

is

<< log T,

which tends to and

b

0

as

(and poles)

O

of

represents here and in the following a funcT ~ ~;

also we assume that

a < a' < b

are the real constants that appear in conditions

in the definition of ~ .

L(s,w,r)

From the Euler product definition of

(A)

L(s,W,r)

and

(C)

and the

functional equation we obtain

L' d T (s,w,r)ds = ~ s log L(s,w,r)

=

=

d log ds

ddslog

~-rd

~ s-h i I I ~-~(s-li)r(--2---)

d + ~ ~(s,#,r)

i=l d

77i=l

~-½(i-s-~i)r(~-=) + ~d ~(s,~,~)

d ½--S + -71-_log f(~,r) ,

where

~l,...,%d are the infinity types that appear in the F-factor

F(l-s,~ ,~);

if we use the Fourier inversion formula we obtain easily

d _h f h(s) d log 77- ~ 2(s-%i) +

(i)

(~) =

-

i=l

d ~ f ~(s) d log 77- ~-½(l-s-Xi)

(~0)

(dim r) log ~½ S ~(s)ds + (dim r) log ~

(~)

i=l S

h(s)ds

(~0)

T

T

= (dim r) log ~½{ / h(½+it)i dt + S h(½+it)i dt + o(i)} -T -T b = 4~i (dim r) (log ~ ) h ( 0 )

From the Euler product

+ o(i).

203

~(x,~,r) = ~ d e t ( l d - p-Sr(gp))-i P

we obtain easily, in the region of absolute convergence

~I

where

X

(s,~,r) = - ~ (log P)x(g~)P n P

is the character of the representation

positive powers of all the prime numbers.

r

-ns

,

and the sum runs over all the

If we observe that

s = 1 + a + it

and

ds = i dt, then we obtain

~,

^

/ h(s)~ (s,~,r)ds

(2)

(~)

f h(s)(- ~ (log p)X(g$)p-nS)ds (~) n P T = - f ~(l+a+it)~ (log p)X(gp)e -(l+a+it)nl°gpidt; -T n P

this last expression becomes, when we substitute the integral defining

h(s),

T - f [ (log p)X(gp)e -(l+a+it)nl°g'p f h(x)e(½+a+it)Xdx idt -T n P T . . n . . . . (½+a+it)u-(l+a+it)nlog p du = - f idt I f (log p)Xkgpjnku)e -T n P T / p -½n (log p)x(g$)h(u)e (½+a+it)(u-n log p) du; = - -T / idt p~

if we make the change of variable

u + u + n log p,

then the last integral becomes

T /

=_ -T / {[nmHP,n P

(u) eitUdu}idt,

204

where

Hp,n(U) = p-½n(log p))<(pg)h(u + log pn)e(½+a)u.

Since

h(u) ~ £ 0

we have

iHp,n(U) I << p-½n(log p) ix(g~)le(½+a)Ue-(½+b)(u+nlogp)

<< (log p) Ix(g~)Ip-n(l+b)e-U(a-b);

similarly we obtain ~v h(s)(- ~ (l-s,~,r))ds

/

O)

C~ 0 )

=-

T _/T{~n ~ HP, n(u)eitudu} idt, P

where

* (u) = p -~n (log p)X(g ~ )h(u - log pn) e (½+a) u Hp,n

where

X

is the character of the contragredient representation

r; again using

the equality T - / dt /H(u)eiUtdu = - 2~H(0) + o(i), -T IR

we collect the results of formulas

i 2~i

(2) and (3)

in the formal identity

^ -½n (log p){x(g~)h(log pn) + ~(g~)h(-log pn)} / h(s)ds 6(s,~,r) = - [ p C(a,T) n P + o(i).

205

The F-factors in

L(s,~,r), contribute, after a simple change of variable,

the formal identity

d

f

:r(:::i>

:og

C(a,T) ½+iT

d

2

d -

½-iT

i=l

-

_ d log~r(l-s-%i)}

i=l

i=l

T d F' ( ½ ( ½ + i t - li) ) + ~ F' ( ½ ( ½ - i t - ~ i ) ) } d t . = ~i / h(½+it) ~ {~ -T i=l

To evaluate the last expression we are lead to consider the following integral

: +iT

d

h(s) 77 :og F(as+b),

2~i ½-iT

where

a

is a positive real number and

change of variable

s = ½ + it

b

is any complex number.

it follows that this last integral is

T i / ~(½+it) 2 i -T

F' a T (a(½ + it)+b)idt

= (2~ a - ~' la,b ) + o(I),

where

h(t)

= h(½+it)

= / h(x)eltXdx, IR

la,b(t) = ~ (½a + b + iat)

and

From the

206

(f,g)

Let us assume

for the moment

+Ia,b(t)

b

is real and put

r' = ~ (½a + b + iat),_la,b(t)

+la,b(t)

then clearly

that

= If(x)g(x)dx.

= _Ia,b(t).

We also have,

r' = ~ (½a + b - iat),

by a well known

formula

([

,

p. 13)

lim +la,b,M(t) +Ia,b (t) = M÷oo

where

+Ia,b,M(t)

: log M + ~ ( t )

M and

+gM(t)

= _ n=0

The Fourier

transformations

of

+~(t)

+gM(x)

= _

n+½a+b+iat

"

is

M ~ n=0

f

-ixt e n+½a+b+iat

dt

and hence

M ixt -k +gM(_X ) = _ [ / e dt; n=O 1R n+~"~a+b+iat

we now make the chan~e of variable

T = at, dT = adt

in the last integral,

obtain ixt ixT/a / e IR/ e ]R nqS~aTb+ia t dt = - a n~a+b+iT

dT ;

to

207

we use the classical distribution formula ([ 27 ], p. 430)

ixT/a a aT= n+½a+b+iT

~f a~

to finally obtain

27 -(n+~a+b)x/a {y e

if if

x> 0 x < O.

Observe that the last formula holds true without any assumption about the complex nature of

b.

Finally

we o b t a i n M

- ~ 2~ e-(n+½a+b)x/a a

+gM(_X ) = { n=0 0

if

x > 0

if

x < O;

therefore we have

0 +gM (x) = { 2~ e(½a+b)x/a l-e (N+l)x/a --• a l_eX/a

The Fourier transform of

+Ia,b,M(t)

if

x > 0

if

x

<

0.

is

+la,b,M(X) = ( 2 ~ 0) log M + +gM(x),

where 2~ 0

60

is Dirac's distribution concentrated at the origin, or equivalently

is the Fourier transform of the constant function

I.

transform of the function

Ia,b,M(t)

= log M + _gM(t),

where

M _gM(t) = _ [ 1 n=on+½a+b-iat

is

_la,b,M(X ) = ( 2 ~ 0 ) l o g M + _gM(x),

Similarly the Fourier

208

where 27 -(½a+b)x/a.l-e -(M+l)x/a , --- e -x/a _gM(x) = { a l-e 0

From the Plancherel

if

x > 0

if

x < O.

formula

<_h ,_la,b ) =

we deduce

<_h ,+la,b> =

= lira (h,_la,b, ~ M->oo e_(½a+b)x/a.l_e-(M+l)x~ = lim {2~h(O) log M - f M+ ~ 0

2__~ a

h (x)d~

l_e-X/a

and

<_h

,

,+la,b >

=

f -~

0 - - e(½a+b)x/a'l-e(M+l)x/a 2~a i -ex/a

= lim {2~h(O)log M - f M÷ ~ 0

2~ e~½a+b)x/a.l-e(M+l)x/a a l_e-X/a

lim {2~h(O)log M M * ~

_

h(x)dx}

•h(-x) dx}.

We have thus proved the following result:

Lemma 6.1. number

a > 0

h

is a function in the class

and a complex number

Wa'b(h)

then we have

If

b

~

and if for a real

we define a distribution

Wa, b

= lim {/ e -(½a+b)x/a l-e-(M+l)x/a h(x)dx - ah(O)log M}, M~ w 0 1-e -x/a

by

209

lim { i / ½+iT fi(s)d M ÷ ~ 2~i ½-iT

We now specialize Lemma (6.1)

i

½+iT^

2~i

log r(as+b) = - Wa, b (h).

to the case under consideration to obtain

d

d

-iT h(s>~s log i : l ~ r ( ~ )

= - i~iw½'-½%i

(h(t)) + o(i)

and

1

½+iT

- 2~---i/

d

h(s) d

l_s_~_i

log ~ F ( ~

½-iT

i=l

i=l ~' "2~i

Let us now go back to the automorphic dimensional complex representation correspond

a distribution

d

) = - ~ W~ _~.~ (h(-t)) + o(i).

and the finite

representation z

r; to the local component

W(~ ,r)

whose value at a function

we make h

in

£

is

given by

W(Voo,r)(h)

co = lim {/ l-e-(M+l)2x d M÷oo 0 l-e -2x "(i=l [ e(%i-½)X)h(x)dx - ½ dh(0)log M}.

Observe that half of the contribution coming from the expression incorporated finite prime W(~p,r)

into the definition of the distribution p

W(~ ,r).

we make correspond to the local component

whose value at a function

W(~p,r)(h)

h

in

~

~

P

(i)

has been

Similarly for a the distribution

is given by

co -½n n = - ~ p X(gp)(l°g p)h(log pn). n=l

We add the local terms

W(~p,r)

corresponding

to all the primes,

including the infinite one, to obtain what in the following is called a Zeta Distribution:

(4)

W(~,r) = ~W(~p,r).

210

The relevant notion that we must now introduce form, which when applied to the distribution W(v,r)

whose value at a function

h

W(~,r)

in the class

W(~,r)~(h(t))

gives a new distribution £

is given by

= W(~,r)(h(-t)).

If we put together the contribution

f ~(s)d (~)

is that of a Weyl Trans-

to the integral

log L(s,~,r)

which comes from the right hand boundary of the contour

C(a,T)

with the contri-

C(a,T),

and if we use the

bution to the integral

f h(s)d log L(l-s,~,r) (aO) which comes from the left hand boundary of the contour obvious fact that the derivative

of

log g(~,r)

is zero and

1 ~U~/o)h(s)dlog f(~,r)½-s = h(O) log f(~,r) + 0(i),

2~i(

then, letting

T + ~,

Theorem 6.2. ~llin

transform;

dimensional

let

we have the following explicit formula:

Let ~

60

be a function in the class £ 0

be an automorphic

complex representation

Assume the Euler product Let

h

L(s,~,r)

be the Dirac distribution,

the formula

(4)

and let

W(v,r) ~

representation

of the dual group satisfies let

CG~

of and

and let

h

be its

G/A and

r

a finite

X

the Working Hypothesis

W(~,r)

be the distribution

be its Weyl transform.

of

§6.1.

defined by

We then have

h(o) = (W(~,r) + W(~,r)~)'h + (~0 log f(~,r))-h, O

its character.

211

where the sum

E runs over all the zeros (and poles) of P is the factor that appears in the functional equation

L(s,~,r)

and

f(~,r)

L(s,~,r) = e (~, r) f (~, r)½-SL(l-s,~,r) .

Remark 6.2.1.

where

y

The sum

Z P

lim

~

T÷°°

I%,t
is the imaginary part of

Remark 6.2.2.

If

is to be understood as a symmetric limit

h(p),

p.

A(s) = ~-½sF(~)~(s)

is the set consisting of the elements

Z p

~(~) =

1

and

is Riemann's Euler product and ~, then the explicit formula

Z W~'h, ~oE~

~(p):O

where

W = ~ Wp,

P co

W "h = - ~ p-½n(log p)h(log pn), P

n=l

~e-½X.l-e-(M+l)2x Woo'h = lim {/ l_e_2X .h(x)dx M ÷~ 0

½h(0) log ~}

and

wl.(h(t)) = W'h(t), ~ ( h ( t ) )

= W'h(-t);

this explicit formula is to be contrasted with the constant term in the Fourier

212

expansion of the Eisenstein series

E(o,s)

for the group

SL20R) and

Foo = {(01 i): n E ZZ}

c(~,s)e (60(s)+p)H(O)

E(d,S) =

+

~* mC~

A(s+l~2.Os(Iml)~.~½sV (2wlmle2OH(d~ 2~imn(O)+pH(~) Iml

where

e pH(d) 0 1 ~(o))( ° e_PH(o))k(0) o = (0

is the Iwasawa decomposition of and~

~,

c(l,s) = i,

is the Weyl group of the Lie algebra

~

algebra of the subgroup of diagonal matrices in

A(s) o(fl,s) = A ( s + l ) '

= ~ SL 2.

fi(s) = -s

which is dual to the Lie Similar such interpretations

can be given for the symmetric term

W(~,r) W

for many Euler products

L(s,~,r)

where there is no ramification.

A deeper understanding of the Weyl symmetries present in the distributions of the form

W(~,r) + W(~,r) ~

and their connection with the constant term of

Eisenstein series, will undoubtedly come from developing explicit formulas for Euler products in several complex variables.

The following lemma, for whose demonstration we refer the reader to the original article of Weil [ 41 ],

Lemma 6.3.

is of some interest.

Let the notation and assumptions be as in Theorem 6.2.

a necessary and sufficient condition for the Euler product

L(s,~,r)

Then

to be entire

213

and for all its zeros

O

to have real part

W(~,r) + W(~,r) ~ + 60 log f(~,r)

Re(p) = ½

is that

be a positive distribution in the sense of

Laurent Schwartz.

Remark 6.2.3.

The proof of this lemma was given by Weil for a type of

distribution defined in the space ~ symmetry;

which did not contain an element of Weyl

the necessary changes that must be made in his argument to apply to the

present situation are of a trivial nature and can be carried out easily.

Remark 6.2.4.

As is well known ([ 34 ], p. 131)

distributions of positive

type have many interesting properties of a hermitian character Th~orSme XVII)

([ 34 ], p. 131,

which are comparable to the Weyl symmetry which is present in the

explicit formula of Theorem 6.2.; for these reasons we believe it is of some interest to study in greater detail the structure of the distribution W(~,r) + W(~,r) ~

and of similar distributions that can be constructed from the

constant term of Eisenstein series in which appear Weyl groups that contain more than 2 elements;

it is also of some interest to look at these results from the

point of view of Bochner's theorem about distributions of positive type.

214

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[3 ]

W. Casselman, On representations of

GL 2

and the arithmetic of modular

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[6 ]

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[8 ]

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[9 ]

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[ii]

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GL(2)

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Gelfand, M. I. Graev and I. I. Piateckii-Shapiro,

Representation

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[15]

H. Jacquet and J. A° Shalika, A non-vanishing theorem for zeta functions of

[16]

GL(2), part II, Springer L. N. 278, 1972.

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H. Jacquet and R. P. Langlands, Automorphic forms on

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215

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GL(2):

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216

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Oscillations of bases in n u m b e r

theory and c o m b i n a t o r i c s

M e l v y n B. N a t h a n s o n D e p a r t m e n t of M a t h e m a t i c s S o u t h e r n Illinois University Carbondale, Illinois 6Z901

I.

Let

]N denote the nonnegative integers, a n d let B ~ ] N

the set of all s u m s then

B

of h

B

order

if h B = IN,

h

order hB

h . If h B

minimality, order

n

B

if B

but for every

e l e m e n t s of

is a m i n i m a l B' ~

B.

b e B

gave e x a m p l e s

of a s y m p t o t i c bases of o r d e r

a s y m p t o t i c basis of o r d e r

Z.

exist a s y m p t o t i c bases

of o r d e r

ISI < 00, then

B\S

B

Similarly,

h ; that is,

there are infinitely

h

contains a m i n i m a l

basis of

that there exist

a s y m p t o t i c bases, a n d N a t h a n s o n [i0] constructed e x a m p l e s for all h_> Z.

basis of

StBhr [IZ] introduced this idea of

and p r o v e d that every basis of o r d e r

h

denote

If h B = IN ,

is a n a s y m p t o t i c basis of o r d e r

such that n ~ h(B\~b}).

a s y m p t o t i c b a s e s of o r d e r

hB

is an a s y m p t o t i c basis of

h . H~[rtter [8] s h o w e d by a nonconstructive a r g u m e n t

minimal

B\S

B

h

contains all sufficiently large n u m b e r s , integers

h . T h e set

but hB' / IN for every p r o p e r subset

a s y m p t o t i c basis of o r d e r

h , but no p r o p e r subset of

many

Let

contains all but finitely m a n y

is an a s y m p t o t i c basis of o r d e r

is a m i n i m a l

.

not necessarily distinct e l e m e n t s of B .

is a basis of o r d e r

I'4, then

B

Introduction

of m i n i m a l

H~rtter [8] and N a t h a n s o n [I0] also Z that do not contain any m i n i m a l

Indeed, E r d B s and N a t h a n s o n [3] s h o w e d that there Z

such that, for e v e r y subset

r e m a i n s an a s y m p t o t i c basis of o r d e r

is no longer a n a s y m p t o t i c basis of o r d e r

basis that does not contain a m i n i m a l

Z.

Z,

Clearly,

S~

but if B

B , if

ISI = o0, then

is an a s y m p t o t i c

a s y m p t o t i c basis.

T h e r e is no classification of m i n i m a l

a s y m p t o t i c bases, nor is there any

s i m p l e criterion to insure that a n a s y m p t o t i c basis contains a minirr~l a s y m p t o t i c basis. minimal of o r d e r

E r d • s a n d N a t h a n s o n [6] p r o v e d that the square-free n u m b e r s a s y m p t o t i c basis of o r d e r Z of s q u a r e - f r e e n u m b e r s

contain a

Z and also that there is an a s y m p t o t i c basis no subset of w h i c h is m i n i m a l .

But it is usually

difficult to d e t e r m i n e w h e t h e r or not a given a s y m p t o t i c basis contains a m i n i m a l a s y m p t o t i c basis.

For example,

the set B

by L a g range's t h e o r e m , but it is not k n o w n basis of o r d e r

Z.

= {mZ+n Z}~,n=0 if B

is a basis of o r d e r

contains a m i n i m a l

asymptotic

Z

218

T h e r e also exist asymptotic bases the sense that, if S ~ B , bu£ an asymptotic 1-minimal

order

nonbasis

bases Let

are

B

If

of order

B

of order

Z

but

= ~

for

every

if

ISI > 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 <

Z

IS] >_ 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 <~(IN),

then

But if there are infinitely m a n y

sets in

I~rdds [I] obtained results ab()ut union bases for

bases and m a x i m a l

h . If h ~

contains a]l

<~(I~)

~(]N)

not belonging to h ~ , h . Recently, D e z a and

analogous to k n o w n results

I shall consider m i n i m a l asymptotic union

asymptotic union nonbases for the collection of finite subsets

of the natural n u m b e r s .

It is easy to prove that every union basis of order

contains a m i n i m a l union basis of order

of order

is called a union basis of

~3 is an asymptotic union basis of

is called an asymptotic union nonbasis of order

about bases in additive n u m b e r theory.

~(~q)

then

and let

not necessarily

h for ~7~(ix[) is contained in a m a x i m a l

h,

h

for

and that every union nonbasis

union nonbasis of order

h.

But

p r o b l e m s about asymptotic union bases and asymptotic union nonbases are m o r e complicated.

For

h > Z there is a trivial construction of m i n i m a l asymptotic

union bases of order

h.

asymptotic union bases.

For

h = Z I shall construct a class of nontrivial m i n i m a l

But it is not true that every asymptotic union basis of

220

order

h

contains a m i n i m a l asymptotic union basis of order

an asymptotic union basis of order for every

h.

I shall construct

Z, no subset of w h i c h is minimal.

h > 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<

b is denoted

the set of all finite subsets of S.

[a,b].

If S ~ ] N ,

then

B. ¢ ~ . 1

Then

h~

for i = i, Z ..... h.

(~.

h~

denote ~

~J ~ J

then the

B I U B Z ~ .. ~ ~h' ... ~ J ~

(h times).

1

consists of all unions of h

Clearly, ~5 Z Z S ~ in

Let

The

c~(S)denotes

If ~ i ~- ~;~(]~q) for i = i, Z ..... h,

union set ~i<3 ~Z<; ... ~ ~h consists of all sets of the form where

Lower

3@C_ ....

Finally, let

IX[

not necessarily distinct sets belonging to ~ .

Let e \ ~ 3

denote the relative complement of

denote the ca rdinality of X.

221

Z.

Minimal

A union basis

~

union nonbasis of order of order

union bases

of order

and maximal h

,~(IN)

for

h for every

Bc ~.

h is m a x i m a l if ~ [.J {A}

union nonbases

is m i n i m a l if ~ \ { B }

is a

Similarly, a union nonbasis

is a union basis of order

h for every

A ~..,~(l~)%k~ . In this section I prove that every union basis contains a m i n i m a l union basis and every union nonbasis is contained in a m a x i m a l union nonbasis. THEOREM union

basis

I.

Every

of order

Proof.

union

basis

of order

h

for

~(~)

contains

a minimal

h.

Let

= {

~

B i}i=l 0o be a union basis of order

h for ,~(IW).

If

is minimal, w e are done.

Otherwise

B.} is a union basis for s o m e B. ~ ~ . 1 1 Let iI be the least subscript such that ~ i =(~\{ Bi I} is a union basis• If ~ l

is minimal, w e are done.

Otherwise, let iZ be the least subscript such that

~2

= ~i ~{ Bi Z} = ~ \ { Bi I' Biz}

~{

is a union basis.

Continue this process inductively.

If it stops after a finite n u m b e r of iterations, then the last union basis in the sequence is minimal. ~

Otherwise, there is an infinite decreasing sequence of bases

~i~2__~

....

where

~k

: ~k-l~{Bik }

and il < iz< i3< ....

is the least integer greater than ik_ 1 such that ~ k _ i k { Bik} I claim that ~':'~:~k°°:l ~ k

:~\{

Bik}k:l

Let X ~ ,~ (IN). T h e set X i
~r'

and

B.. ~i~

Since

~r

h~*.

Thus,

If ~ * \ { B i} is a basis for s o m e and ~k_iX{Bi} Therefore,

is a union basis•

~ *\{B.}

THEOREM

Z.

Proof. h~.

Let

Let ~

~'"

tj ~i k for all k <_ r since

Btj~X

. Therefore,

is a union basis oforder

B.i ~ ~ * '

is a nonbasis for e v e r y

Btj ~ ~ * h.

then ik_ 1 < i < ik for s o m e

]B. ~ ~:"~,

a n d so ~3;:" is a m i n i m a l

i

h for ,Y~(JN). Every union nonbasis of order

in a m a x i m a l union nonbasis of order

X/

i such that r is abasis, thereare sets Btl,

But this contradicts the minimality of ik.

1

union basis of order

h.

is finite, so there is an integer

and also t.] / i k for all k > r since and so X ~

ik

is a union basis.

is a minimal union basis of order

~ r such that X = ~ hj:l Bt'" J Clearly,

for j :i ..... h,

and

is contained

h.

be a union nonbasis of order

~i = ~ ~

h for .~(IN)

(~(]N)\,~(X)).

Clearly,

h.

Choose

X ~ .~(IN) with

~ 1 is a nonbasis since

k,

222

X~

h ~ I.

so

~

l

But ~ I

contains all but a finite n u m b e r of elements of ~(~q),

is contained in a m a x i m a l union nonbasis of order

h.

and

223

3.

M i n i m a l asymptotic union bases

A n asymptotic union basis there are infinitely m a n y

(~

of order

h

is m i n i m a l if for every

sets in ..~ (Eq) that do not belong to h ( ~ \ { S} ). A n

asymptotic union basis of order

1 is simply a co-finite subset of .~(Eq).

no m i n i m a l asymptotic union basis of order

1 exists.

there do exist m i n i m a l asymptotic union bases of order that every asymptotic union basis of order union basis of order

h.

h

3.

h > 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]<+l] / ~,

(iii) If Ph
B Z =[n k + i, nk+l]

(iv) If IR.kl > I, then ~k+1

or

for every

B c (~k+l;

then X ~ Z(~k+1 u

{P~k});

BI, B z ~ ~9k+l<J.~([l, nk]),

B Z = P~k and

B 1 = [nk + I, nk+l];

does not contain every nonempty subset of

[nk + 1, nk+l].

Proof.

Case~: Suppose

i : [ % + 1, % + j I suchthat

}Jl = m}.) the f o r m where

Since Izl : % + i - %->

Z~ =.)~(I)\{~,I}.

If lit = Zm,

fix x ~

3, there is

s'V

I, where

S' ~ S ,

1%' ~ 1%, S' C S , and hence nk+l]

2

a family

of subsets of

: {JC_I I I__< IJ I _<m}.

= { J ~ I I i__< IJl < m -

l } V {J_~IIx{ J and

and, second, those ofthe form

J E ~.

~k+l

Clearly,

for any J ~ #

JI, Jz ~ ~

. Since

and 1%Z~ 1%. Then

s'U

J,

U {I}

satisfies conditions (i) and (iv).

with XI = X(-~I / ~ . Let i%' = X(~it and S' = X(-]S.

then X : R U (S' U I) ~ Z ( ~ k + i V {%})

R'U

I( (~k+ 1 and X ( - ~ I ~

X = Itl V S I <_) X I. Suppose that X I : I. Then

Z(~k+iU

and

Let (~k+l consist of the following two types of sets: first, those of

Let X ~ [ l ,

some

Let R : ~ , S : [ I , % ] \ R ,

(If Ill = 2 m + i, let ~

I andlet ~

for all X ~ ~k+l'

Then

I~I>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 <~,

Let XC[I,_ nk+1]

then

with

X ~ Z(~k+l<-) {R-k])~- Z(~{~) elements of ,2~(IN). Thus,

g whenever

[~I <°° . This proves

227

4.

M a x i m a l asymptotic union nonbases

A n asymptotic union nonbasis an asymptotic union basis of order union nonbases of order

(~

of order

h for every

S ec~(Eq)\~.

is

T h e asymptotic

1 are precisely those 03 C-,~(]N) such that o~(]N)\~ is

infinite. T h e r e is clearly no such m a x i m a l union nonbasis of order nonbases of orders

h is m a x i m a l if B~J {S}

~,

hence no m a x i m a l asymptotic

i. It is not k n o w n if there exist m a x i m a l asymptotic union

h > Z.

But it is possible to construct asymptotic union nonbases

that cannot be e m b e d d e d in m a x i m a l asymptotic union nonbases. THEOIREM

6.

then for every k > l Proof. S ~ h~

If

If

there are only finitely m a n y ~

sets X ~ h ~

sets X ~ ,2~(~) with

also contain all but finitely m a n y COROLLARY.

Let

~

sets X ¢ j~(]N) with

m a x i m a l asymptotic union nonbasis of order

where

~

is

Sup posethat X ~

implies that h ~ X ¢ h~ Ch~

nonbasis of order

basis of order S, T { ( ~ ,

h~

must

.~

h

such

cannot be e m b e d d e d in a

h.

~

IXl >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
Constructing

now a multiplicative function

n =

we see that

~(m,n) = [ { ~(pr,pS)

where

m = { { pr , P

P p , P (2.7)

~ t(m,~) ~(~,n) = 6m, n

since the left hand side of (2.7) arises by multiplying together the left hand sides of (2.4) for all When

p.

t(m,n)

Thus

~(m,n) = t*(m,n)

which is therefore multiplieative.

is multiplicative as well as normal lower triangular (2.1) and

(2.2) take the forms

(2.8)

x

m

=

t(m,d) Yd

dim

and (2.8')

Ym =

~ t*(m,d) x ddlm

This generalizes the usual inversion formulae. We have, of course, also the dual set of formulae:

if

t(m~n)

is not

assumed multiplicative but is normal lower triangular and if the sequences Ym

xm

and

are connected by the relations

(2.9)

Xn = ~ t(m,n)

Ym'

m

then we have that

(2.9')

Yn = ~

t*(m,n) x mm

Here, since the sums on the right hand side are infinite, one has to assume that, say, the

Ym

are such that the suma occurring converge absolutely.

instance, is the case if we assume that the vanish for

(2.10)

then

m

sufficiently large°

xd =

Ym

This, for

(and as a consequence also the

For multiplicative

~ t(m,d) Ym' dlm

t(m,n)

we get that if

xm )

236

(2.10')

Yd = ~ t*(m,d) x m. dlm We call a symmetric function

(2.11)

f(mgn)

Q(x) =

satisfies

Q > 0

positive definite if the quadratic form

~ f(m,n) x m x n m~ n

for all real sequences

xm

with at least one and at most finitely

many non-zero elements. For

f(m~n)

positive definite, we can always find functions

normal lower triangular

t(m,n)

(2.12)

g(n)

and

such that

f(m,n) = ~ g(~) t(m,~) t(n,~),

these functions are uniquely determined and can be expressed rationally in terms of the

f(m,n). If we, in addition, require

that both and

g(n)

~(pr, pS)

and

t(m,n)

and

to be multiplieative~ it is easily seen

will also be multiplicative *). Namely we define

~(pr)

by the relations

f(pr pS) = ~ ( p t ) ~(pr p t ) 7 ( p S pt) for all t and ~(pr pS) = 0 for s > r. For each p this determines

r,s ___ O, t(p r ,pr) = i, ~(pr)

f(m,n)

~(pr, pS)

uniquely for all

r,s > O.

We now construct the multiplicative functions Writing

m = I I pr

n = I [ pS

we then have

~(m)

and ~{(m,n).

f(m,n) = I If(P r'ps) =

1 1 ~(Pt) V(pr,pt) ~(pS,p t = ~(~)~(m,~)t(n,~). Thus ~ p t identical with g and t which are therefore multiplicative. For multiplicative positive definite

and

t

must be

f(m~n) (2.12) therefore assumes the

form

(2.13)

f(m,n) =

~ g(d) t(m,d) t(n,d). dlm dln

3. conditions

Suppose that we wish to determine the minimum of xn = 0

for

n > 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
xn = ~

t*(m,n) Ym"

(3.2) gives (3.4)

m In particular the condition

(3.5)

xI = I

takes the form

~ t*(m,l) Ym = i. m
The minimum is then by standard procedures found to be

%

(3.6)

.in

t * (m~l) 2

~

'

g(m) and the minimizing

xn

are given by

(3.7)

Xn = Qmin " ~ t (m,n)* t*(m~l) m
Finally, if we assume t(m,n)

and

t (m,n).

f(m,n)

to be multiplicative,

from the

or not, depending on the nature of the

f(pr, pS)

for

r,s ~ O.

This may be simple

f(pr, pS).

We shall consider an application of the preceding to a sieve problem.

Let us denote by

prl]n that

exact prime divisor of

prI~

and

designate a set of

~(pr)

assume that for fixed are disjoint if n

r ~ s.

,p

= i,

we say that

p

is an

n.

Assume now that we have a set of primes

integer

g(m),

These functions have to be determined by computing the

g(pr), t(pr,pS) and t*(pr,p s)

4o

then so are

p

p.

residue classes modulo p

For each r

the residue classes removed We introduce the notation

lies in one of the

w(p r)

p

r

with

r > 0

we

to be removed or excluded. modulo p nX(p r)

r

and

modulo p

s

to denote that the

residue classes excluded

modulo pr.

We

We

238

define

nX(1)

whenever

for all integers

n,

and further for

nX(d)

if

nX(p r)

prlldo

Let there now be given a set of integers associated Xd = 0

d > 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
f(pr,pS) = 0

for

(r-s)rs ~ 0,

240

g(pr) = (e(pr-l) . @(pr)) 8(pr) 6(pr-l) "

(4.6) For

r > s~

we find e(pr-l)_@(pr),

if

s = 0

e(pr'l)'e(pr), @(pS)

if

0 < s < r .

t(pr'pS) = I "

(4.7)

Similarly for

r > 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
Thus

*r (

t (p ~i) 2 g(pr)

(4.9)

i 6(pr)

i £(pr-l)

1

Thus the minimum of

f(d,d') Xd Xd' is i d
e(p r)

e(p ~'I)

From (4.4') we thus get the upper bound of the number of integers that the sieve leaves in

I

as X

X+Z

(4.10)

2

prlld e

e<

A result~ identical in form with (4.10)~ has been given earlier by Gallagher and Johnsen

*)

, who derived it from the large sieve and under the rather restrictive

assumption that for each

*)See Gallagher [2].

r > i

the

m(pr)

residue classes excluded modulo p r

are

241

equidistributed among the

pr-I @(pr-l)

residue classes that remain modulo pr-I

after the earlier exclusions modulo p,o..,p instance~ that P P o @(pr) e(p r'l)

w(Pr) r-i i) 9(pr-

r-i

This restriction implies, for

is always an integer (or~ otherwise expressed 9

is always an integer) *) .

References Io

Ao Selberg:

Remarks on sieves°

Proceedings of the 1972 Number Theory

Conference~ University of Colorado~ 1972~ pp. 205-216. 2o

P. Xo Gallagher:

Sieving by prime powers°

Proceedings of the 1972 Number

Theory Conferenc% University of Colorado~ 1972~ pp. 95-99°

*)When I first saw Gallagher's paper [2], I thought that the unnecessary restriction was due to faulty technique in the use of the large sieve inequality. As I~ however~ was unable to derive (4.10) from the large sieve without imposing the same restriction, it remains an open question whether the more general result given here is implied by the large sieve inequality or not.