Hecke operators and Euler products

Hecke operators and Euler products

Citation for published version (APA):

van Lint, J. H. (1957). Hecke operators and Euler products. Utrecht University.

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,)

НЕСКЕ

OPERATORS

AND

EULER PRODUCTS

PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE WIS- EN NATUURKUNDE AAN DE R!JKSUNIVERSITEIT ТЕ UTRECHT, ОР GEZAG VAN DE RECTOR MAGNIFICUS Dr. L. SEEKLES. HOOGLERAAR IN DE FACUL TEIT DER DIERGENEES-KUNDE. VOLGENS BESLUIT VAN DE SENAAT DER UNIVERSITEIT, TEGEN DE BEDENКINGEN VAN DE FACULTEIT DER WIS- EN NATUURKUNDE ТЕ VERDEDIGEN ОР MAANDAG 28 OKTOBER 1957 ТЕ

4 UUR PRECIES

DOOR

JACOBUS HENDRICUS V AN LINT

GEBOREN ТЕ BANDOENG

~о-~ ~i г~~:

!1\ ·. . ,: 01...~ I • · - ' - '

---

-.... f ~ ·. r:. . .п '· .. ~> !

7

h.r~ р ~о

1 "

1

'

~-- '-·' -~~-~--_:_-~-- -c----,.-J

1

Т.Н.

Eii'HJ!:ovr::N

\

\

_______

---·-

---··

PROMOTOR:

§ О. Introduction.

The Fourier coefficients of modular f9rms have been studied

Ьу many mathematicians. Although many interesting results have

Ьееn found, still enough proЬlems remain to keep us interested in this subject. One of the powerful methods used in studying the coefficients is the theory of Hecke~s operators Тп (Hecke [1]). Hecke considered modular forms of integral dimension for

con-g~uence groups Г (N). The operators Тп are linear comhinations

·

(ат

+

Ь) (а Ь)

·

of transformed functions

f .

d , where О d runs through а complete system of in regard1 to Г (N) nonequivalent transfor-mation matrices of order п. For instance, for modular functions of dimension - l' and step 1:

f

1

Тп

=

п'-

1 2:

t(ат

d

Ь)

d-'.

ad=n

bmodd,d>O

The operators шар the set of integral modular forms of dimen-sion - l' and step 1 into itse]f. These operators were studied

thoroughly (Hecke [lJ, Petersson [2]). It was proved that under certain conditions modular forms could Ье written as linear

com-Ьinations of eigenfunctions of these operators. со

If f(т)

=

2: а (n) е lГ is eigenfunction of all Тп, the associated

п=О

00

Dirichlet series ер (s) = 2: а (п) п-s can Ье written as an Euler

n=l

product П (1 - a(p)p-s

+

s(p) pk-I-2s)-1. р

То find analogous results for modu1ar forms of non-integral dimension, operators of the type of Тп would have to Ье defined. Hecke [2] considered the fшictions J18 (т) 5-Ь(т)

where а+ Ь is an odd integer, and proved that operators ТР could generally not

А general theory of "Hecke-operators" was given Ьу \Vohl-fahrt [1]. The operators he defined, and some theorems concern-ing these operators, will Ье given in § l. Now that we have these operators, we are of course interested in the results that can Ье found Ьу applying them. W ohlfahrt considered applications to the functions {} (т, 9, stl, Pk), defined Ьу Schoeneberg [1]. This gave formulae concerning the number of representations of an integer Ьу quadratic forms in an odd number of variaЬles.

For many years the coefficients of the powers of ;i (т) have

been studied, and special attention has been given to the findiDg of coefficients which are _ zero. The results concerning these coef-ficients, that have been found Ьу Newman [1, 2, 3, 4] and Rade-macher [1] suggest that they can Ье proved and generalized Ьу applying Hecke-operators to the functions ;i1 (т). This is done in

§ 4 of this thesis after the necessary operators have been defined in § 2 and § 3.

Wohlfahrt proved that all Hecke-operators could Ье built up out of the operators he defined excep_t if the transformed func-tions occuring in these operators were linearly dependent; in which case more operators could Ье defined. Sometimes the ope-rators can on]y Ье defined if there is linear dependence, as is the case for the functions ;ia (т) .Э-Ъ(т), studied Ьу Hecke. In § 5 we

shaB prove some theorems about this linear dependence, and find all integral modu!ar forms for the groups Г (1_) and Г {} for which this linear dependence occurs, under the condition that 2r is an integer, if r represents the dimension of the modular forms.

§ 1. Definitions and notation.

1.1 Г (1) is the modular group, i.e. the group of unimodular two-rowed matrices (;

~)

with rational integral elements. _ Г (N) is the subgroup of matrices for which а= d - 1 (mod N) and Ь =с= О (mod N).

· Г ₀ (N) is the subgroup of matrices for which с

=

О (mod N).

r

0_{(N) is the subgroup of matrices for which Ь =О (mod N).}

Га.

is the subgroup generated

Ьу (~ ~)

and

(~ -~).

7

We shall write

(~ ~)

=

!,

(~ ~)

=

U and

(~ -~)

=

Т.

Let Г Ье а subgroup of finite index of Г (1), r а real number. We shall call а function f(т), defined for lm т >О, а modular form of dimension - r and with multiplier system v for the group

Г if it satisfies the following conditions (cf. Petersson [1 ]) : . 1. ln every closed region in the halfplane Im т >О, f(т) has as only singularities, at most а finite number of poles.

2. For L

= (;

~) е Г

and lm

т

>

О,

the following relation ho1ds:

(

ат+ ь)·

f(Lт)

=

f ст+ d

=

v(L) (ст+ d)' f(т), (1.01) where v(L) depends only on L and J v(L) J

=

1. (For v(L) we

shall also write v (;

~)).

Here

(ст+

d)' is defined as follows: For (с, d) =/=(О, О) and lm т >О, (ст+ d)'

=

e'10g(c•+d), where that branch of log(cт

+

d) is taken, for which log(cт

+

d) is real if ст

+

d

>

О, с

f=.

О. And for с = О we define:

. . sgnd-1

(ст

+

d)'

=

d' ~

1d1'

е

-

m r -2- .

If

М

=

(:~

::)

and S . (;

~)

are rational unimodular

ma-trices and MS

=

М' =(то: тз:).

the number r;(M, 5)

=

r;(г)(M,S)

т1 т2

is defined in the following way:

'(т 'т

+

т ')'

(m1 Sт

+

т2)'

=

r;(M, S). (ст+ d); .

Then r; (М. S) = е2_{п;"', where}

11 is an integer which does not depend

on т. For the multipliers the following relation holds:

(1.02) 3. ln the closvre of а fundamental region of Г, f(т) has as only possiЬle singularities, poles in the various uniformizing variaЬles. These poles do not cluster about r~al parabolic limit points of the fundamental region. So we have, if UN е Г and we write v ( UN) ~ e'lnix :

1: 2niт:n

2niN:-t со N

f(т)

=

е

2:

а(п) е ; (1.03)

n=-k

!Г, -r, v! will denote the class of modular forn:is for the group

Г, with dimension - r and multiplier system v. We call the class trivial if f(т)

=

О is the only modular form in the class.

If f(т) is а modular form of dimension - r apd S

=

(с

а

db)

а

rational matrix with

1S1

>

О, we shall write:

f(т)

1

S

=

JSlr

(ст+ d)-r f(Sт). (1.04)

We then have:

(1.05)

If f(т) е !Г, -r, vj. and

Q

is а transformation matrix, we have:

fj

Q

е jГ J

Q.

- r , v J

Qj.

(1.06)

where ГI

Q=

Г(I) 11 Q-1гQ and

v/

Q is the multiplier system

determined Ьу :

v(L) J Q

=

rr(~~g~i)Q)

v(QLQ-1) for L

е Г

J Q. (1.07)

For further theorems on modu!ar forms of non~integral dimension, see Petersson [1].

We shall call а matrix Q primitive if the greatest common divisor of its elements is 1. If 1

Q

1

=

п; we say

Q

is of order п.

Some modular forms we shal1 consider are the following:

ni-i ni'rn.2

12

=

·=

(12)

-а. 11(т)

=

е П (1 - e2

"'1n•) =

2: -

е .12 (Im т>О). (1.08)

n=l n=I п ·

ii(т) is а modular form for Г(l), of dimension

-t. There· are

several expressions for the multiplier system (cf. Rademacher [2]). In the interior of the upper т~halfplane ii'(т) is free from poles and zeros, and ii (т) vanishes at т =О and т = ico.

ь.

2(~)

=

'1 2

З-(т)

=

1

+

2

L

e"'i•n2

=

.

n=l ii(т+ 1) ( 1.09)

9

In' the fundamental region of Г {}• {} (т) has no poles. The Ье~

haviour in т

=

-1 can Ье found from (1.09) to Ье as follows:

2;n;i ( 7: ) ( 7: )

( +

т 1)1. •..;т=.е С.( ) В ;+'! ~ ,:::...,Спе 2ni ~ п

п=·о (1.1

О) с. The Eisenstein series:

1

Gk(т)

=

2,(k) m~:2 (т1т

+

m2)-k

=

(27Гi)k

= .

- 1 _L "\" rr. · (,,.,\ ,,.2;n;i<n (1 11)

- ' 1 ((k).(k--,-l)Jn~l-к-,-1\••1- ' ,."., where 11', (п)

=

L:

d', and k is an even integer

?: 4, are

d/n, d> О

modular forms е jГ(l), -k, Ч· Every modular form е jГ(l),-k,

1J

k

can Ье writteri uniquely as

L:

спдn(т)Gk-₁₂,;(т), (O::s;п::s;U,

п

k - 12п-=/::. 2), where G₀

=

1, and д(т) = 1124_{(т) (cf. Hecke [1]).} 1.2 W е shall now give some definitions and theorems from Wohlfahrt's general theory of Hecke~operators (Wohlfahrt [1]).

lf Г_{2 is} а subgroup of finite index of Г₁, we write jГ₁ : Г2! for а

complete system of representatives of а decomposition оfГ₁ into

left cosets of Г₂• •

Theorem 1. If К= jГ; -r, v! and Л

=

j&, -r, v*j асе two classes о{ modular forms,

Q

is. а trCJ.nsformation matrix witb v \

Q

=

v* оп В п Г \ Q, and & п Г 1 Q is о{ finite index in Г(f), the operator

T~(Q) defined Ьу

fl

T~(Q)

=

L:

v*(V)-1

fl QI

_{V (1.12)}

v

е

ie :

е Г\ г 1

Q!

maps К into Л.

Remark: If Q*

=

п Q-1 _{· where} п

=

_{1 Q 1. and} Т~ _{( Q) is} de~

fined, then

~

(Q*)

сап

also

Ье

defined.

Theorem 2. We define а Hecke-operator to Ье ап operator which trans{orms tlie modular forms { о{ а class jГ, -r, vi into forms о{

j&,

-r, v*i· the operator being а linear comЬination {with coetfi.cients independent о{ {) о{ transformed {uпctions f J Q with

primitive Q о{ а {ixed order. Тhеп all Hecke-operators defined {or the whole class jГ, -r, vi сап Ье built ар out о{ the opera-tors defined in theorem 1 with the following exception : Let

Q

rип through а complete system о{ primitive transformatioп ma-trices of fixed order, which are left-inequivaleпt with respect to

Г. lf for all f of jГ, ~r, v! the transformed functions

fl

Q

are

liпearlg dependeпt, other Hecke-operators thaп those meпtioned

above сап Ье defiпed.

Remark: It is also of interest to consider those functions of jГ, -r, v! for which linear relations for the

fl

Q exist. For these functions other Hecke-operators can Ье defined. Of course these operators are not necessarily Hecke-operators for the other tions of the class. Functions for which this holds are the func-tions 11 (т), 11 3 (т), .Э-(т) and 114_{(т) ,Э--J (т), studied Ьij Hecke [2]. We} shall give more examples in § 5.

Sometimes the operators defined in theorem l can Ье written in а form which is easier to handle. Consider К= jГ, -r,

v! ,

л

=

j&.

-r, v*I· Let .

г

(N)

~ г о

& and let

т~

(

Q)

Ье

defined and (п, N)

=

1, where п

= [

Q[.

Then the system:

LR

=

92~

=

jR:

R

=

(~

b1;).ad

=

п,

а> О, Ь mod d, (а, Ь, d)

=

Ч (1.14) is а complete system of in regard to Г(l) left-inequivalent, pri-mitive transformation matrices of order n.

Theorem 3. lf 92 ~ Г Q& апd [& () Г(l) Q: & Г\ Г [ Q]

=

1 we have

fl

T~(Q)

=

L

л(R)-

1

fl

R.

Re

LR

( 1.15) Here л(R)

=

15(М, Q) 15(MQ, V) v(M) v*(V) i{ R

=

MQV with М е Г, V е &. We call this the normal form о{ Т~ (Q).

/:с.

11

ln defining the operators т~

(

Q) the main difficulty is proviпg that v*

=

v 1

Q.

In fact if К is given and а is а given group it

is possiЬle that for а no multiplier system v* exists so that

v*

=

v 1 Q and

je,

-r, v*! is non~trivial. For certain groups we

shall study the multiplier systems in § 2, апd then define some special operators of the type т~

(

Q) in § 3.

§ 2. Some properties of multiplier systems.

We shail consider multiplier systems for the groups Г ( i),

го

(2) and

Г.9" estaЬlishing

relations between

v(;p

~)

and

v (:

ь~)

.

First we remark that for all groups Г with -,/ е Г, we have:

l' ( - / )

==·

e-:л;ir • (2.01)

2.1 The case Г

=

Г{l). Г(l) is generated Ьу U and Т for which the following relations hold: Т2

₌

_{- I and (UT)}3

₌

_-I. From these we find for Л

=

v(U) and µ,

=

v(T) the relations:

л

=

~

₆

е 6

and µ,

=

л -з

=

±

е 2 • .(2.02)

Here ~₆ is а б~th root of unity. We see that for Г(l) only 6 different multiplier systems are possiЬle (for every r).

In the following we shall use the fact that for every М е Г(l)

there is an integer w (М), independent of r, so that:

v(M)

=

лw(М). (2.03)

Proof: Expressing М in the generators U and Т, we find for

v(M) а product of factors rт(М;, Mj) and powers of Л and µ,.

All rт(М;, Mj) are powers of е2_{"';', therefore powers of л. Using}

,и.,-:--- л-з we find the result (2.03).

If f(т) е !Г{l), -r, v!, application of (1.06) gives:

For

f(т)

=

'11(т)

and

р

>

3

а

prime, v*

(:р ~)

=

в(d)

= ( :) ,

where (:) is the quadratic restsymbol (Hecke [2]).

This gives us the relation :

eТJ!w(;Ь~)-p.w{;P~}!

- (:) • _(2.05)

Using f(т)

=

'113 _{(т) we find:}

1 ,.1 \

\;).

(2.06)

We now prove:

Theorem 4. If !Г(l), -r, v! is non~trivial, r is ап integer and

р

>

3 is

а

prime, then v (;

Ь~)

=

vP

(;р ~)

апd

vP ( ;

Ь~)

· v

(;Р ~).

Proof: (2.05) · implies · w (;

Ь~)

-

р.

w

(;Р ~) =·О

(mod 12) and as in this case Л12

=

_{eZяir - 1, the first equation is proved.}

Т

aking the

p~th

powers Of both sides We find VP ( ;

Ь~)

=

vP

2

(;р

;).

From this the second equation follows because

р2

=

_{1 (mod 24) and all multipliers are 12~th roots of unity. If} 3/r and л

=

±

е 6

, these results are also true for р

=

3, which

can Ье proved using (2.06).

Corollary: if f(т) е !Г(l), - r ,

v!, r

an integer, then

f(т) е !Г (12), -r. Ч·

(

а ь)

·

(а ь) (Ь -а)

·

Proof: If с d е Г(12) we have µ,.v с d :._ v d -с. Now (d, 6)

=

1 so we can apply theorem 1.

13 We find, using Л12

=

_µ,4

=

_1:

~)

=

!

vva-d(~ -а:)

= µ,d = µ, ( _-1

Ь

_-с ad)

=

µ,

-Зd

=

µ, (d >О) (d <О).

From this follows

v(;

~)

= 1 if (;

~) е Г{12).

Theorem 5~ lf, in jГ(l), -r, v!, 2r = l is ап odd integer, and

р

>

3 is

а

prime, then v (;

Ь~)

= (:)

vP

(;р ~)

and

v

(:р ~)

=

= ( : )

оР

(;

Ь~)·

Лi(/-1) Лi :Щ

у _1_2_ 12 12 6

Proof: Jn this case л

=

'>б. е е

=

~₁₂ е where ~

₆

=

у12

= '>12= 1.

The first equation now follows from (2.05) and the second сап Ье proved iп the same way as in theorem 4, if we remark that now all multipliers are 24~th roots of uпity. As in theorem 4,

the result is also true for

р

=

3 if 3/l and

л

=

±

е

6

.

Applying theorem 4 or theorem 5 twice, we find:

Theorem 6.

lf,

in jГ(l), -r, v!, 2r is ап integer, and р

>

3 is

.

(а Ьр2

) (а·

а przme, then о с d

=

v ср2

~).

Again, this is also true for

р

=

3 if 3/2r and

л

=

+е

6

.

.

(ар2 Ь)

.

(а Ь

)

From these equations, others, e.g. v. с d

=

v с dp₂ сап

Ье proved easily {2r an integer). With these theorems the multi~ pliers can easily Ье determined iп those cases where (с, 6)

=

1

or (d, 6)

=

1.

We shall поw prove analogous results for the groups Г0(2) and Г1J.. First we remark that, Ьу (1.06), if f(т) е jГ₀{2), -r, о!, then:

2.2 The case Г

=

Г₀(2). Г₀(2) is generated Ьу U and Т*

=

=

(~ ~)·

for which the relation (U-1

Т*)

2

=

-1 holds. Writing v(U)

=

;..* and v(T*)

=

µ,*, we have:

-:тtir

*

+

*

2

-µ, =_Ле . _(2.08)

Using (2.08) we find that for every М е Г₀(2) there are num-bers w₁ (М) and w2 (М). independent of r, so that:

("){\О\

\ • ... V . / f

Considering the multiplier systems of 11 (т) and 11 (2т), we now find two relations :

е ~

![

WJ (:

Ь~) -

2w2

(~ Ъ~)

] -

р

[

wl

ер ~)

-

2w2

ер ~)

]

!

=

(~

) . (2.1

О)

е ~

![

2wl (:

Ь~)

- w2 (:

Ь~)

] -

Р

[ 2wl

ер ~)

-

"'2

(:Р ~)

]

!

= ( : ) •

(2.11)

Applying these formulae in the same way as was done for

Г(l), we find:

Theorem 7. If we consider only multiplier systems in which ;.,*

is а 24-th root of unity ( and therefoтe all multipliers are 24-th roots of unity), the theorems 4, 5 and 6 hold if Г ( 1) is replaced

Ьу Г₀(2).

2.3 The case г

=

г ff· г ff is generated Ьу U2 _{and Т. W е write}

v(U2)

=

л*

and v(T) = µ,

=

+

е~.

The calculations in this

(

т

+

1)

case are rather tedious. We use the functions 11

-2- and &(т). W е shall only state the result:

Theorem 8. If we consider only mUltiplier systems in which J..

*

is а 24-th root of unity, the theorems 4, 5 and 6 hold if Г(l) is replaced Ьу Г {}·

15

§

3. Some special operators T~(Q).

In this paragraph we shall consider operators T~(Q) with

Q

=

(6

~). (~ ~)

or

(6

2

~)·

If

possiЬle,

we shall write these operators in the normal form (1.15). In the following we have К= jГ₁, -r, v!, Л

=

jГ₂, -r, v*!· l

=

2r. Furthermore р will always represent а prime number larger than 3.

3.1

Г

1

=

Г

2

=

Г(l),

r

ап

integer, Q

=

(6

~)·

Ву (2.03) the multiplier systems are determined Ьу л

=

v (И) 2ni~

nir-and л*

=

v*(U). If we have л-= е 6 е 6 , we choose k2

=

k2 nir

r(p- 1) 2ni6

=

pk1

+

₂ (mod 6). Then л *

=

е е

6

satisfies л

*

=

;..,Р,

and so Ьу (2.03) and theorem 4 we have : v*

(;Р ~)

=

vP

(;Р ~)

=

v (;

Ь~)·

Therefore v*

=

v 1 Q on Г0 (р)

=

Г2 r\ Г1 1 Q.

Both conditions in theorem 1 are satisfied. Hence we can de~

fine

Т~

( Q}. (Remark:

(~

( Q) can also

Ье

defined for these classes). In vlew of Iater applications we choose N

=

24 to write Т~ (Q)

in the normal form (see ( 1.15) ). The conditions of theorem 3 are satisfied. C'R_ consists of the р

+

1 matrices

(

Р О) _О

d (1

24Ь) · 1 an О р , (Ь

=

О, .... р - 1). (3.01) We find:

j

.

1 л (R)

=

r{p-1) (- 1) 2 for

R

=

(~ ~)·

(₀1

2Р4Ь).

for

R

=

(3.02)

We shal1 write Т(р) for the operator we have defined. We found: When f(т) е

!

Г(I), -r, vj. r an integer, then

where

r(p-1) р-1

f(r)IT(p)

=

p'f(pr)+(-1) 2

2:

f(r+

24

Ъ).

(3.03)

Ь=О Р

If 3/r and л =

±

е 6 we can define Т(З):

f(r) \'

Т(З)

=

3'

f(Зr)

+ (- 1)'

~

t(r +

3

ВЬ)·

(3.04)

Ь=О

If 1 = 2r is an odd integer, we cannot define an operator Т(р) as was done above for Г_{1 =} Г₂ = Г(l).

Proof: А necessary condition is v * = v

1

Q, therefore л *

=

ЛР.

(

аЬ)

-

(а Ъ)

Thenv*=vPforaH с d е Г(l). Wethenhavefor ер d Е Г₀{р):

v*

(:р ~)

= v (:

Ъ~)

= (

~)

vP

(;р ~),

and this is equal to

vP

(;р ~)

only if (:) = 1, therefore not for

allшatrices

in

Г

0

{р).

We see that operators Т(р) can only Ье defined for these classes in the exceptional case mentioned in theorem 2 {see § 5).

3.2

Г

1

=

Г

2

=

Г{l),

r

ал

integer, Q =

(~

2

~).

W е now take v*

=

v. Then Ьу theorem 6: v* = v 1 Q on

Г₀(р2_{) = Г2 Г) Г1 1 Q. The conditions of theorem 1 are satisfied.}

То write T~(Q), which operator we shall write as Т(р2), in the normal form, we take :

СУ)

=

{(р

₀

2

о

₁

)

••

(Ро 21Рь)

-{L with (Ь = 1, .... р-1); (О Р1- 24Ь) 2 w1th

.

(Ь -

_

О, •.•• р -2 1) . } (3.05) We find: л(R)

=

r(p-1) (-1)_2_

(

р

2

О)

(1

24Ь)

for 0 1 and 0 р2 f or

(

Р 24Ь) 0 р . (3.06)

17

So we have: if f(т) Е !Г(l), -r, v!, r an integer, then

f(т) 1 Т(р2) Е !Г(l),

-r, v!

where f(r) 1 Т(р2)

=

r(v-1J р-1 (

+

24Ь) Р2-1 (

+

24Ь) =р2_'f(р2_т)+(-1)-2_-Р'

_L

_f

_{рт ___}

₊

_L

_f

_~-

_·

Ь= 1 Р Ь=О Р (3.07)

lf we denote f(т) 1 T(p)I Т(р) Ьу f(т) 1 Т(р)2 _{the following relation}

holds:

r (р-1)

f(т)

1

Т(р)

2

=f('Г)1 Т(р

2

)

+

(-1)_2_

р'(р

+

1)

f(т).

(3.08)

~ir

If 3/r and Л =

±

е 6 we can define Т(9) Ьу replacing 24 Ьу 8

in (3.07). Then (3.08) holds for р

=

3.

For

Г

1

=

Г

2

=

Г(l),

r an integer, Q =

(~ ~)

where(n,6)= 1,

we can define an operator Т(п) in the same way Т(р) was de~ fined in 3.1. In the case v* = v = 1, these operators сап Ье

compared with Hecke's operators Тп (Hecke [1]). Wohlfart [1]

found the relation

Тп

=

п-

1

L:

h'

т(~

2

).

h>O,h2/n

For the operators Т,(п) we can find а multiplication theorem. Wohifahrt [1] proved that: .

Т (т) Т(п)

=

Т (тп) if (m, п) = 1. (3.09) In the same way as for Т(р) and Т(р2_{) we find the normal form:}

k-1 = pkr { (рkт)

-+.

L

v=l Р" - 1 r (р" - l) ( k-v

+

24Ь

L

(-1)--2-.p(k-v)r.{ р Т

)+

Ь=1 Р (Ь,р)= 1 pk-1 r(pk-1)

+

L

(-1)-2 -

t(T

+

24Ь) • (3.10) Ь=О pk

This gives us for k

>

2 :

r(p-1)

Combining (3.08), (3.09) and (3.11) we find:

r {р-1)

~

T(n}

=

П

( 1 - (-1)-2-pr-2s ) • (3.12}

п• r(p-1)

(п,6)=1 р>З l -T(p).p-•+(-l)-2-pr+I-2s

Here the series and the product are to Ье considered f ormally only.

3.3

Г

1

=

Г

2

=

Г(i},

l

=

2r

ап

odd integer, Q =

(6

2

~)·

As in 3.2 we take v*

=

v and Ьу the same reasoning as is . used there, we find that тji (Q) can Ье defined. We shall also

call this operator Т(р2_{). For CR, we choose (3.05) and find:}

1 for

R

=

(~

2

~)

and

R

=

(Ь

2

~).

л(R)= _. ( 1>

1

(

3

Ь)

when

р

=

1 (mod 4)

j

(3.13) ~ (24Ь) Р е 2 -

=

р

i1

(

3

Ь)

_when

р

=

_{3 (mod 4)} \

Р

. fo,

R

=

(~

2:).

So we have: If f(т) е jГ(l), -r,

vj.

l

=

2r an odd integer, then

f(т)IT(p2_{) е jГ(l), -r, vj. where:} f(т) 1 Т(р2₎

=

=

р2'fр2т)+'УрР'

PL: 1

(3b)t(pт+24b)+P

2

L:

1

t(т+24Ь)

b=I Р Р Ь=О Р (3.14)

with '/'р

=

1 when р

=

1 (mod 4) and '/'р

=

i-z when р

=

3

(mod 4).

For 3/r,

л

=

+

е

6

,

we can define

Т(9):

f(т) 1 Т(9)

=

=

32r

f(32т)

+ i-z 3,

±

(~) f(Зт

+

8Ь)

+

~ f(т

+

ВЬ).

Ь=О 3 3 Ь=О 9

~~~~~---·--··- -·---··

19

We can define T(n2_{) for all п with (n, 6)}

=

_{1 in the same way.}

We find: f(т) J T(p2k)

=

p2kr f(p2kт)

+

2k-1

+

2:

pv-1 -nir(pv-1)

(24b)v

(р2k-ит

+

24Ь)

2:

е 2 - • p(2k-v)г

f

,

v .

+

Ь=О р Р v=l P2k -l (т

+

24Ь)

+

2:

f

2k • (3.16) Ь=О Р This gives us : Т(р2) Т(р2)

=

Т(р4)

+

p2r+I(p

+

1), (3.17) and for k

>

1: T(p2k) Т(р2)

=

Т(р2н2)

+

Р2,+2 T(p2k-2). (3.18) Fi:om (3.16), (3.17) and T(n2} T(m2)

=

T(n2m2} if (n, m}

=

1 we find: T(n2) _ ( 1 _ p2r+ 1-11 ) (n,

lr

=

1 -;;;-

р ~

3 1 -

Т(р2)

p-s

+

p2r+2-2s • (3.19)

(Series and product are to Ье considered formally).

3.4

Г

1

=

Г

2

=

Г

0

(2)

or

Г{),

2r

=

1

ап

odd-integer, Q

=

(6

2

~).

The reasoning is the same as in 3.3. We now use theorem 7 and theorem 8. In the case Г₀(2), we take (3.05) for Сfг, finding

(

Р2 о)

(1

24Ь)

.

Л(R)

=

1 for

R

=

0 1 and

R

=

0 р2 , wh1le fqr

R

=

=

(~

2

~)

we find: '(R)

--!

(~)

when

р

=

1 (mod 4) " if л*12

₌

_-1, i1 ( 3 :) when

р

=

3 (mod 4) (3.20)

j

(

6 : ) when

р

=

1 (mod 4) л(R)

=

6Ь · i1

(-р)

when

р

=

3 (mod 4) if л*12

₌

_1.

We find:

л(R)

=

1 for

R

=

(~

2

~)

=

(~

4

~)

we have

(1

48Ь) and

R

=

0 р2 , while for

R ____:_

~ (~)

л

(R)

= (

ii

(6:)

when р

=

1 (mod 4), when р

=

3 (mod 4).

For the Т~ ( Q) we find formulae analogous to (3.14).

(3.21)

(1

Ро).

3.5 Г₁

=

Г(l), Г₂

=

Г₀(р), 2r

=

l ап odd integer, Q

=

0

,

In 3.1 we proved that if l is an odd integer, we cannot de~

fine

Т(р)

as in (3.03). The

ope~ator T~(Q)

we now define,

re~

sem.Ыes Т(р) very much and is therefore of interest. We shall call this operator Т₀ (р).

For ( ;

~) е Го (р),

we define v*

с~ ~)

= (:)

vP

с~ ~)

=

=

v (;

7)·

Ву

(1.06) this is

а

multiplier system for

Г

0

(р).

ln fact we have: if

f

(т) е

j

Г (1), -r, v

! ,

then f(рт) е

j

Г₀ (р), -r, v*

! .

Furthermore we have v*

=

vl Q on Г~ (р)

=

Г

2

п Г

1

1 Q. There~

fore-T~(Q)

=

Т

0

(р) can Ье defined. In this case we cannot find а normal form.

For

jг

0

(р): Г~ (р)!

we take

(~

2

;ь). (Ь =О,

" ..

р-1).

We then have: If f(т) е

j

Г (1), -r, v

! ,

2r

=

1 is an odd integer and v* is the multiplier system of f(рт), theп f(т) 1 Т₀(р) е

j

Г₀(р), -r, v*j. where

f

(т)

1

То (р)

_:__

~

1

f ("

+

24

Ь)

,

Ь=О р (3.22) 2

("+8Ь)

.

~ir

and f(т)I Т0 (3) == ь~о

f

3 if 3/r and Л

=

±

е . (3.23)

21

Remark : As was said in ( 1.13), we can also define

Т~(

Q

*) in the case considered here. After some calculations we find :

g(т)IT~(Q*)

=

r2 _

1 ( 0)

r -

1 (о

_ )

=

ь~о g(т)[ 2~Ь

1

+

µ.-i

ь~о g(т)[

1

24Ь~

· (3.24) An interesting property is : If f(т)e jГ(l),-c,v!, !~ (\\ v then f(т)[ ~~ ~)ITЛ(Q*)

=

f(т)JT(p2). (3.25) 3.6

Г

1

=

Г(l), Г

2

=

r

0_(p),

2с

=

_l

ап

_{odd integec, Q}

₌

(~ ~)·

This case is suggested Ьу the previous one. It is а Ьit more difficult in the calculations. That Т~

(

Q) can Ье defined is proved in exactly the same way as was done in 3.5. We sha11 call the operator Т0 _(р). ( 1 о· For jr0_(p):

гg(р)!

_{we choose}

24

Ь

1),

(Ь =О,

.•..

р-1).

This gives us :

f(т)IT~(Q)

=

р'f(рт)

+

р' :~

1

1

t(

₂₄

:Ь~"+

₁

).(24Ьт

+

1)-r. Let Ь', ·d' Ье integers, so that 242_ЬЬ'

=

_{1 (mod р) and pd'}

+

+

242_ЬЬ'

₌

_1. Then we have :

p'.f(24b~T+

1 ).

(24Ьт

+

1)-r

=

~

Р

("

+

24Ь')

-

24Ь']

=

р'.

{ (

р+

2

-!Ь')

(24Ьт

+

1)-r = 24Ь т

+

d' р _ ( -24Ь') -n~(p-I) (т

+

24Ь') - . . е

.f

.

р р

So we have: If f(т) е

!

Г (1), -r, v

! ,

2r

=

l is an odd integer and v* is the multiplier system of

r(;).

then f(т) 1 Т0 (р) е

!

Го (р), -r, v*

! ,

where f(т) 1 Т0(р)

=

(-24) -.п;,(р-l) p -l ( Ъ) (т

+

24Ь)

=

p•f(p т)

+ - -

е 2

L

-

f . Р Ь= 1 Р Р (3.26)

The connection between {3.22) and (3.26) is given Ьу the fo1-lowing relation :

f(т) 1Т₀(р)1 Т

=

µ, f (т) 1 Т0 _{(р). (3.27)} The operators we have defined are of interest because of their applications which will Ье discussed in § 4.

§ 4. Applications;

We shall now discuss some applications of the operators de-fined in § 3, especially to powers of )j (т). Many of the resu] ts

we find, have already been found Ьу М. Newman (cf. Newman

[1], [3], [4]) and Н. Rademacher (cf. Rademacher [1]). The in" teresting fact is here, that а11 these resuits are consequences of the existence of Hecke"operators.

If f(т) is an integral modular form е !Г (1 ), -r, v!, where 2r is an integer, we see from (1.03) and (2.02) that the Fourier ех" pansion of f( т) has the form :

00 2ni-rn

L

а(п) е 24 • (4.01)

п=О

00 2:тti'rn·

We can a1so write f(т) as

2:

а'(п) е N , with N/24 and а'(п) =О,

п=О if (п, N)

>

1.

The Dirich1et series connected with this Fourier series is: со

q> (s) =

2:

а (n) п-s. (4.02)

n=l

If, in {4.01) we have (п, 24)

=

k for all п with а(п)

:;t:

О, we

shall write : _{- со .}

q>(s)

=

2:

a(nk) п-s. (4.03)

23

For the first applications we consider:

4.1 Even powers of 11 (т).

If l= 2r is even, then 11l(т) е !Г(l), -r, vj. where v is determined

Ьу л

=

е

6 , r an integer.

First we remark that these functions could also Ье discussed with Hecke's theory. applied to !Г(12), -r, lj (cf. theorem 4,

corollary). The results would not follow as easily as they do now. In this case we find for Hecke's operator ТР, the relation:

r(p-1)

Tr

=

(-1)_2_ Р-1 Т(р). (4.04)

Now consider f(т) 1 Т(р) е !Г(l). -r, v*j, where v* is determined

лirp Ьу л*

=

еб. Ву (3.03) we have: f(т) 1 Т(р)

=

00 2ni~np r(p-1) р-1 00 2л;(<+24Ь) п

=

р'

L

а(п) е---и'

+

(-1)_2_

L

L

а(п) е 24Р n=O Ь=Оп=О

=

n~O{pr.a(;)

+

(-l)<r;l)

р.а(пр)} /~Zn.

(4.05)

(As usual, we define а(х)

=

О if х is not an integer.) If 3/1, formula (4.05) is a1so true for р

=

3.

We are interested in knowing if the Dirichlet series rp (s), con~

nected with 11l (т), are Euler products in regard to certain primes. ln the next paragraph we shall prove: If

f(

т) is eigenfunction for Т(р) with eigenvalue с, then:

r(p-1) t:p(s)=(

L

а(п)п-s) (1-(-1)-2-c.p-s-I+ п =!=О (mod р) We now prove : r(p-1)

+ (

- 1, - 2 -} р r-1-2s'-1 ) • (4.06)

Theorem 9. lf О< 1~24, l is even and 1 (р - 1) =О (mod 24),

then 11l(т) is eigenfunction о/ Т(р).

Proof: If l(p-1)=0 (mod24), we have ЬуЗ.1: л*=J.and

11i (т) 1 Т(р) . .

therefore g(т) = '11l(т) е jГ(l), О, Ч· Usшg the propert1es of

'11 (т) stated in § 1, we see that g(т) has no poles in the funda~

mental region of Г {l) except possiЬly in т

=

i

=·

Ii;i the Fou-rier expansion of 11l(т) 1 Т(р) we have for the first coefficient which

is not zero: п = l (mod 24) and therefore п?: l. Hence g(т) has no pole in т = i со and so g ( т) has no poles at а11. W е now use the fact that an automorphic function

f

е jГ, О, 1

!·

which has no poles in the fundamental region of Г, is а constant (cf. L. R. Ford, Automorphic Functions, page 94). We have proved that g(т) is а constant. It is easily seen that this constant is р. a(lp)

if we write: 11l(т)

=

~ а(п) е 24. (We shall use this

О

<

п l (mod 24)

notation for the rest of this paragraph.) We have proved:

11i (т) 1 Т(р)

=

=

р. а (lp). '11l (т) if [ is even, [ (р - 1)

=

О (mod 24). (4.07) Theorem 10. Define [1

=

pl - 24

[f

~].

If l is even,

О<

l

~

24

and [₁

> [,

then '11l (т) is eigenfunction о{ Т(р).

In this case v* ~ v, and therefore the eigenvalue сап only

Ье О.

Remark: theorem 2 of Newman [3] is contained in this theorem. Proof: In this case

g(т)

=

'11l(т~l/(~(p) Е jГ{l),O,v*.v-1i,

(л*. л-1₎6 _{= e"'ir(p-l)}

=

_{1. Hence g}6_{(т) е jГ(l), О, 1}

!·

g6 _{(т) has no poles in the fundamental region of Г (1) except}

possiЬly in т

=

i

=.

In the Fourier expansion of 11l (т) 1 Т(р),

3""&i?:l1

the first coefficient which is not zero is the coefficient of е 24, and as [₁

>

l we find that g6 _{(т) has а zero in т = i со. Ве~} cause g6 _{(т) has no poles in the fundamental region of Г (1), g}6 _(т)

-must Ье а constant. Therefore we have g (т)

=

О. We have

found : If l is even,

О

<

l

~

24 and -pl - 24

[~~]

>

!, then

25

Theorem 11. !{ О

<

l::::::; 24, l even, then 11i (т) is eigenfunction о{ Т (р2_{) for all р}

>

_{3, and {or р}

=

_{3 if 3/ l.}

Proof: This theorem is proved in the same way as theorem 9 was proved.

From theorems 9 and 1 О we find that 111 (т) is eigenfunction of Т(р) for all р

>

3 if l = 2, 4, 6, 8, 12 or 24, i.e. if l is an even divisor of 24. Denoting the Dirichlet series connected with .

111 (т) Ьу lfz (s) we have ф2 \/ ")\ - тr { 1 - 'р\ -- s 1 ( - 1 \ - - 2s \ -1 1 ~ - l l \ 1 - ., ] 2 \ } • f! -г- 1 - - ; f! } р>З ' р

q;4

(s)

=

П (1 _ 'Тб (р). p-s

+

p1-2s)-1 р>З (4.09) (4.10) ~б (s) = fi

(i -

T4(p).p-s+·(-l\p2-2s1-J р>2 р ! J (4.11) о/

₈

(s)

=

П (1 - т

3

(р). p-s

+

p3-2s)-1 р>З ~

₁₂

(s)

=

П (1 - т

2

(р). p-s

+

p5-2s)-1 р>2 ~

₂₄

(s} = П (1 - т (р). p-s

+

pll -2s)-1 р (4.12) (4.13) (4.14)

For the coefficier1ts тп (р), the following expressions are known

,

(cf. Schoeneberg [2], [3]). Let 7r1 Ье а Gaussian prime, 7r1 its

con-jugate. ,

i

з ,3 ( ) - 71"1

+

7r1 'Т3 р - if р

=

7r17r1 and 7r1

-

1 (mod 1/-3) 'Т5 (р)

=

о if р

=!=

1 (mod 3) if р

=

7r17r1 , and 7r1

-

1 (mod 1/-4) if р

=!=

1 (mod 4)

71"1

+

71"1₁ if р

=

_7r17r'1 and X(7r₁₎

=

1, where Xis

the character mod 2 1/ - 3 of order 6, for which

Х

(/:;)

=

е

-;"'.

These expressions and an analogous expression for т₁₂ (р) are given in Schoeneberg [3].

Ву (1.08) we have т₁₂ (р)

р

=

1 (mod 12), р

=/=

1 (mod 12).

Expressions for т₂ {р) and т (р) are given in Schoeneberg [2]. (Remark: for some of the functions we have discussed, operators

Т(2) and Т(3) can also Ье defined as we see from the products (4.09) to (4.14).)

We now discuss the even powers of '1 (т) which are not eigen~

function for all Т(р), (1~24). From Hecke's theory of the Тр (Hecke [1]) and the formula (4.04) we see that 111 (т)

=

L:

f;(т},

i

where the modular forms f; (т) are eigenfunctions of all Т (р). We can take the f; (т) liвearly independent. Applying Т (р) we find ii1_{(т)1 Т(р)}

=

_L:

_{с;(р)f;(т), where the с;(р) are the eigenvalues.}

i

Applying Т (р) once more we find

'11_{(т)1Т(р)1 Т (р)}

=

_L:

_с;2 _{(р) f; (т).} Ву (3.08) and theorem 11 :

111_{(т)1 Т (р) 1 Т (р)}

₌

r(p-1) i

=

c(p)ii1_(т)

+

_{(-1)_}2_{_pr(p+} l)ii1(т)

₌

c'(p}ii1(т).

Because the f; (т) are linearly independent, we have:

с;2 (р)

=

с' (р) for all i.

Taking the functions {; (т) with с; (р)

=

V

с' (р) and those with С; (р)

=

-V

с' (р) together, we find: '11 (т) is the sum of 2 functions which are eigenfunction for all Т(р). We refer to

this as (4.15)

W е now consider 1110 ( т). This is an eigenfunction of Т (р) for

р

=/=

5 (mod 12) Ьу theorems 9 and 10. We write ii 10(т)1 Т (р)

=

00 2niтn

=

L:

Ь (п) е 24 _{• For р}

=

_{5 (mod 12) we find Ь (О)}

=

п =О

27

From this we see that the function g (т)

=

=

J1

10

(:~i(~(p) е fГ(l),-4,1!

has по poles in the fundaшental region of Г (1 ). Hence g (т) must Ье а multiple of G4 (т). Comparing coefficients we find:

J110_(т)IT(p)

=

_{р.а(2р)G4(т)J1}2_{(т) for р}

=

_{5 (mod 12). (4.16)} The two functions {; (т) шentioned in (4.15) are linear com~ Ьinations of J110 _{(т) and}

_G

4 (т) J12 (т). If we write

00 2пiтп

G

₄ (т) J12 _(т)

₌

_2:

_{с(п) е} 24 _{, we have for р}

=

₅

_{(mod 12), p-f:-5:}

n=O

G4 (т) 112 (т) 1 Т (р) = р • с (1 Ор) J110 (т). If J110 (т) + х G4 (т) 112 (т)

is eigenfunction for all р then р. а (2р) = а2 р. с (lOp).

For the decomposition of J110 _{(т) into eigenfunctions of all}

Т(р) we find:

J110_{(т) =tj11}10_{(т) + i8 G4(т)11}2_{(т)! +tj11}10_(т)-41_{8 G4(т)11}2_(т)!. (4.17) The eigenvalues are:

р. а (lOp) for р

=

1 (mod 12),

48 р. а (2р) and - 48 р. а (2р) resp. for р

=

5 (mod 12), О for р

=

7 (mod 12) and р . 11 (mod 12). The Oirichlet series connected with the two functions, have Euler products. ComЬining these we find with (4.17):

1

rp1a(s)=- П (l-p4-2s)-I. П (1-a(l0p).p-s+P4-2s)-1.

96 р-7,11 р=1 (mod 12) (mod 12)

!

p-s

П (1 - 48 а (2р) . р-•

+

p4-2s)-I (mod 12) - fl (1

+

48а(2р).р-•

+

p4-2s)-I

!

р_5 . . (4.18) (mod 12)

Another proЬlem we are interested in, is finding the zeros of

00 00

Pz(n) where П (1 - хпу

=

2:

Pz(n)xn.

=

Writing '11i (т)

=

:2:

а{п) е 24 we have Pz(n)

=

а(24п

+

l).

п=О

ТаЫеs of the pz (п) were given Ьу Newman (Newman [1]). If

1/24, the zeros of Р1 (п) are found easily from the well known

Euler products (4.9) to (4.14). For l

=

10 we have Ьу (4.18):

р₁₀ (п) is zero if at least one of the primes

=

7 or 11 (mod 12) divides 24п

+

10 in an odd power, and also if all primes

=

5 (mod 12) divide 24п

+

10 in even powers. This last condition does not give us any zeros not covered Ьу the first condition. W е shall now consider '11 14 (т). Ву theorems 9 and 1 О this is an eigenfunction of Т(р) for р

=

1 (mod 12) and р - 5 {mod 12).

'11 14 (т) is also eigenfunction of Т (Р) for р

=

11 (mod 12). '1114_(т)1Тfр)

Proof: g (т)

=

₁₁₁₀ (т)' is а modular form е

1

Г (1), -2, 1

j

without poles in the fundamental region of Г (1). The only

function satisfying these conditions is g (т)

=

О. We remark that this same proof can Ье applied in the case of '11 25 (т) and

р

=

11 (mod 12).

Ву the same same method as was used for 1110 (т) we find:

'11 14 (т) / Т(р)

=

-р.а (2р) G₆(т)'112_{(т) for р}

=

_{7 (mod 12). (4.19)}

'111;(т)

=

t

j'1114(т)

+

3б;V3 G5(т)'112(т)

!

+

i

+

t

j '1114 (т) - 360V3 Gб (т)'112(т)

j

(4.20)

=

fi

(т)

+ {

₂ (т) with {; (т) eigenfunction for all Т (р), (р

>

3). From this we find ~

14

(s) as sum of two Euler products:

i4(s)

=

72oivз

_!I

(1

+(-1(;1рб-2s)-1

Р=5. 11 (mod 12) П (l -a(l4p)p-•+p6_{-2s)- 1.} p:=l (mod 12)

{

pg.?

(1

+

360iV3.a(2p)p-•) - p5- 2•)-l (mod 12)

p:g

7 (1 - 360i vз

.

а (2p)p-s - р6-2•)-1 ~ • (mod 12) (4.21)

29

So р14 (п)

=

О if one of the primes

=

5 or 11 (mod 12) divides 24п

+

14 in an odd power; or if all primes

=

7 (mod 12) divide 24п

+

14 in even powers. Again this second condition does not give us any zeros not covered Ьу the first condition. То discuss 1116_{(т) completely we must introduce an operator} Т(2). То do this we choose Л

=

j

Г (1). -8, v*

! ,

where v* is the multiplier system of 11 8 (т). Ву using one of the kпоwп ех~

plicit expressions for the multiplier system of 11(т), it is easily

sееп

that we have v*

=

v 1 Q if Q

= (

~ ~) .

Therefore

Т(2)

сап Ье defined. In the same way as (4.16) апd (4.19) we now

fiпd:

1116_{(т)I Т(2) = 2 G4(т)11}8_(т),

and for all primes р

=

2 (mod 3) the analogous resu]t:

1116_(т)

₁

_{Т(р) = р. а (8р)}

_G

4 (т) 118(т).

(4.22)

(4.23)

Ву theorem 9, 1116_{(т) is eigenfunctioп of Т(р) for р}

=

_{1 (mod 3).} The decomposition into eigenfuпctions of all Т(р) is:

1116 (т) 1 (f";5(s)= П (l-a(lбp)p-s+P7-2s)-l. 12VIO р-1 (mod3)

{

p_2(mod3) п (1 - 6 v10. a(8p)p-s + p7-2s)-1 (4.24)

-

п (1 +6VIO.a(8p)p-s+p1_- 2s)- 1}. (4.25) р=2 (mod3)

From (4.25) we fiпd: р16 (п)

=

О if all primes

=

2 (mod 3) divide

3п

+

2 iп even powers, Ьut this is not possiЬle. In this way we do not find zeros of р₁₆ (п). lt is поt known if р₁₆ (п) has zeros. Ву the taЬles in Newman [1], р₁₆(п) ~О for п::::;:; 400.

The fuпctions 1118 _{(т), 1120 (т) and 11}22_{(т) сап also Ье treated as} was done for the other even powers of 11 (т). For 1122 (т) the for~

mulae Ьесоmе much more complicated. We shall give the results for 1118 (т) and 11 20 (т). 11 18(т)IТ(р)=р.а(l8р)1118_(т) for

v=

1 (mod 4), ( 4.26) 1118_{(т) 1 Т(р) = -р. а(бр) G} 6(т)116_{(т) for р}

=

_{3 (mod 4).}

i ;8 (s) = п (1 - а {18р) p-s + p8-2s)-1. 48V 35 р = 1 (mod 4)

{

p=3(mod4) п (1 + 24iV35 а{6р) p-s - ps-2s)-1 П {1 - 24iV 35 а(6р). p-s - p8-2s)- 1}. '(4.27) р =3(mod4)

As was the case for '116_{(т), this formula gives us no zeros of} Р1в (п).

'120(т)1 Т(р) = р. а (20р) '120 (т) for р

=

1 {mod 6),

(4.28) '120_{(т)IТ(р)=р.а(4р)G8(т)'1}4_{(т) for р=5 {mod 6).}

- 1 . ff₂₀(s) = П (1-a(20p)p-s+P9-2s)-I. 576V70 р = 1 (mod б) { _Р=5 П _(modб) (1 - 288V70 а (4р). p-s + p9-2s)-1 - П (1 + 288V70 а (4р) .p-s + p9-2s)-1_{}. (4.29)} p=5(modб) We find no zeros of р₂₀(п).

4.2 Odd, positive powers of '1 (т).

Consider '1i ( т) е jГ (1 ), - r,

v!

where 2r

=

l is odd, v deter~

nir

mined Ьу Л

=

еб. As we have seen, operators Т(р) generally cannot Ье defined for these functions. W е write

00 2ni_?:n

'1~(т)

=

f(т)

=

L

а (п) е ---U-.

n=l

We find for р

>

3 from (3.13) for every f(т) е jГ(I), -r,

v!:

00 2nirnp2 f(т)IT(p2_)=p2_'

_L

_{a(n)e---U- +} n=l · 00 (р-1

(

3Ь) 2nibn) 2ni~n

+

rP·P'

L

а(п)

L

-

е-Р- е2Г

+

n= l b=I Р оо (р2-1 2nibn) 2"'i~n

+

L

а(п)

L

е р2 , е24р2

=

n=l Ь=О =

~

{p2r

а

(--;-)

+ (-

1

)z;t.

(Зп )рг+t а(п)

+

р2

a(np2)}

е zп;_;п.

n=l Р Р Р (4.30)

3i

If 3/l we have f(т) 1 Т(32₎

₌

n~l {з

2

_{'а:( ~)}

_{+ ( -}

_{1 {-;}1 •

(n~З)

3r+t a(n)

+

9a(9n)}e

2

~:n.

(4.31)

Theorem 12. If О

<

l

<

24, l odd, then 111 (т) is eigenfunction о{ ,Т(р2_{) for all р}

>

_{3, and for р}

₌

_{3 if 3/1.}

111 _{(т) 1 Т(р}2_{) ·} Proof: Consider g (т)

=

₁₁₁ (т) . Ву the same reasoning

i:1

- " v -w--:;" -...., ·u·""'u.J ....,.._ 1'v-- -·--- --·"---J.: t::Vt::ll _l.JUWt::L" U.L ~& "(,.,.\ ' i \ • j t ~,~ ~~~ +J..~+ VV~ .;J"'-~ LJ..J.U\. 'tf\"J""' (.._ ,-, (,.,-\ с Sr \..1.Jt'-'t (1 \ () 11 А~

and that g(т) has no poles in the fundamental region of Г(l).

Therefore g(т) is а constant. Ву computing g(т) in т

=

i

=

we find 1-1

( ) / - 1 \

- 2

(3

l ) 1 2 f ~) g т

= \

р}

.

р р'+•

+

р а ,lp'- for р

>

3 and 1-1 (l/3) g(т)

= (-

1) 2 _•

_{3 .}

_3'+.!

+

_{9а(91) for р}

₌

_{3 if 3/1.}

With this theorem we can deduce relations for the numbers

pz(n). We have

(

п)

·(-1)

1 ; 1

(3n)

,

pZr а pi

+

Р

.

р

.

р'+•. a(n)

+

p2_.a(np2 )

-

.

.

=

j( /

)';'(~

1

). ,,,.Н

+

р'. a(lp~} а (п).

р2-1 We now write п = 24т

+

1, др= 24

and we use a(n)=p

(

п 24 -

l)

•

Then ( 4.32) becomes:

(

т -lд) р2". pz р2 Р

+

(4.32)

{ (- l)l-l(

3 ) r+_!_[(24m

+

l) ( l )]

}

+ р 2 р .р z р . - р.

-

p2.pz(lдp) pz(m)+

+

p2_.p 1(mp2

+

[др)= О for р>3. (4.33)

For instance: l

=

5, р

=

5, т

=

5п gives us

(

п -

1)

53_р5

-5-

+

бр5(5п)

+

р5(125п

+

5) =О. In the same way: l

=

5, р

=

7, т =}п

+

3 gives us

(

п -

1)

73_р5

-7- - 1бр5 (7п

+

3)

+

р5(343п

+

157)

=

О. These, and many other relations are mentioned in Newman [1]. As we have seen they are all special cases of the general for~

mula (4.33).

For 3/1,

р

=

3, formula (4.33) is true if we replace

(__!__)

Ьу

р

(

1

_~

3

_).

The author did not succeed in finding zeros of pz(n)

Ьу

using these formulae as was done for even l. Of course if а zero is known, others сап Ье found' from (4.32). We have Ьу induc~ tion: If а (п)

=

О and. р2

/ п (р

>

3) thert а (np 2k)

=

О.

If 3/1, а (п)

=

О and 27 / п then а (32

k n)

=

О. lt has been shown

that р_{15 (53)} =О (cf. Newman [1]). So we find:

(

п2k_

1)

р₁₅ 53п2_k

+

₁₅

24

=

О for all odd п. (4.34) Whether there are other zeros than these and whether the fact that р_{15 (53)}

=

О has а deeper significance or not remains an unsolved proЬlem.

The formulae we can find Ьу applying the operators Т₀ (р)

and Т0 _{(р) are generally too complicated to Ье of any use in} finding relations for the coefficients pz (п). То illustrate this, we shall give one example (here the result is still relatively simple).

• 115(т) 1 Т0 (7) Cons1der g(т) =

115

(

7

т) . Ву (3.22) we have: 00 2niт:n 7

L:

а(7п) е 24

g(т)

=

:0-0 _ _

2~.:гс;~,п-.7 е jГо(р), О,

11.

L

а(п)е 24 n=O

33

Applying the operator Т and using (3.27) we find :

2л:i п · -2:п;i 49п

_7з ~

(.!!_)

~ (п) е-1

... • 24

+

75

~

a(n)

е---;;;;-·

24 п=О 7 п=О g(т)

=

=

L

а(п) е~· 24 -2ni n · п=О

W е see that the function g (т) - 73 _{has а pole of order 1 in}

-2ni

e2_{"'i" in т}

=

_i

=

_{and а zero of order 1 in е} 7_{" in т}

₌

_{О and по}

other poles or zeros in the fundamental region of Г₀(р). The

{

11 (т)

}4

function

11 (

7

т) has the same properties and therefore we have

g

(т)

- 73

₌

с. {

11

11

_(~;)}

4

•

Comparing coeffici~nts we find с

=

7 а (77)

=

70, an d so:

115_{(т)I Т0(7) = 7011}4_(т)11(7т)

+

₇3₁₁5_{(7т). (4.35)}

Even this simple result does not give us linear relations for the

р5 (п).

4.3 Applications to 11- 1 (т).

The operators we defined сап also Ье used for negative values of r. An interesting case is the function 11-1_(т).

-:тti-C ()О

11-I(т) = e---i-2

L

р(п} e2"'inт' п=О

where р(О)

=

1 and р(п) is the number of partitions of п. Ву proving formulae as 11-1_{(т) 1 Т0(5)}

=

₅2₁₁5_(5т)11-6_{(т) and}

11-1_(т)

₁

_Т

0(7)

=

72₁₁3_(7т)11-4_(т)

+

₇3₁₁7_(7т)11-8_(т),

we can give proofs of the Ramanujan congruences

р(5п

+

4) =О (mod 5) and р(7п

+

5) =О (mod 7) and others.

These proofs were given Ьу Rademacher (Rademacher [1]). The

theorems he used were special cases or applications of the the~ orems we have proved.

We could also apply the operators Т(р2_{) to 11-}1_{(т). As 11-}1_(т)

is not eigenfunction of these operators, the results do not give us linear relations for р(п).

We give one example:

( . =

11-1(т) 1 Т(р'1)

{r(1)·

_р2-1

1}

g 'Т) J1-Р2(т) е ' 2 '

and has no poles in the fundamentaJ region of Г(l). Therefore

g(т)

can

Ье

written as

п~О xnдn(т)G

12

{k-n)(r), (k=P

2

2

~

1 ). . 11-1_{(т)JТ(25) -17250 . .} 1 For шstance

11

_25(т) 691 д(r)

+

0G12(т). 4.i Applications: to

_{- -}

.Э-( т). .

We can aiso use· the·· methods of this:paragraph to prove some results of Hurwitz [l] and Sandham [1] on the representation of

а square as а sum of squares. We apply Т(р2_{) to .Э-Z(т), (l'> О,}

. { 1

1

l odd). We find .Э-Z(т~J Т(р2) е Гff.

-z·

Vfff• where Vff is.the multi-plier system of · .Э-(т) .

.Э-Z(т) 1 Т(р2_{) has no poles in the fundamental region of Гff.}

b.e-cause .Э-(т) has this property. Therefore the behaviour in т

=

-1

is described Ьу

:

(т

+

1)1;21.э-z (т)

1

Т(р2)!

=

e2;i

С:1)

z

~-·

cv'. e2"'i

С:1)"

.

'P=-k l wh·ere k ~

8

"

(4.36)

.

,э.z (т) 1 Т(р2₎ Let l Ье 3, 5 or 7. Cons1der g(r) = ,Э.l(т)

We have: g(т)'е jГff, О, Ч· As if(т) has no zeros in the funda-mental region of Гff except in the limit points т

= ±

1, we see that g ( т) has no poles in the: fiшdamental regioп of Г {} except

possiЬly in т

=

±J.

From (1.10) and (4:36) we see that g(т) is regular in т =

±

1. Therefore g (т) has no poles in the funda-mental region of Г {}· Hence g (т) is а constant.

Le1 r1(n) Ье· the number of representations· of п as· а sum of

00

l squares. Then ifl(т)

=

2:<

rz(n) е"iтп and