Two problems related to the non-vanishing of $L (1, \chi )$

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

P

AOLO

C

ODECÀ

R

OBERTO

D

VORNICICH

U

MBERTO

Z

ANNIER

Two problems related to the non-vanishing of L(1, χ)

Journal de Théorie des Nombres de Bordeaux, tome

10, n

o

_{1 (1998),}

p. 49-64

<http://www.numdam.org/item?id=JTNB_1998__10_1_49_0>

© Université Bordeaux 1, 1998, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Two

_problems

related

to

the

_{non-vanishing}

of L (1,~)

par PAOLO

CODECA,

ROBERTO

DVORNICICH,

et UMBERTO ZANNIER

RESUME. Dans cet

_article,

nous étudions deux

problemes

à

_priori

assez

_{éloignés, l’un}

se _rapportant à la

_géométrie

_{diophantienne,}

et l’autre à

_l’analyse

de Fourier. Tous deux induisent en realite

des

_questions

très

_proches,

relatives à l’étude du _rang de matrices

dont les coefficients sont nuls ou

_égaux

à

_{( (xy/q) ), (0}

_{~ x, y}

_q),

ou

_((x))

= x -

_{[x] - 1/2}

_désigne

la

_partie

fractionnaire "centrée" de x. L’étude de ces _rangs est liée au

_problème

d’annulation des

fonctions L de Dirichlet au _point s =1.

ABSTRACT. We

study

two rather different

_problems,

one

_arising

from

_Diophantine

_geometry and one

_arising

from Fourier

_analysis,

which lead to _very similar questions,

namely

to the

_study

of the ranks of matices with entries either zero or

_((xy/q)),

0 x, y q,

where

_{((u)) = u - [u] - 1/2}

denotes the "centered" fractional

part of x. These

ranks,

in _turn, are

_closely

connected with the

non-vanishing

of the Dirichlet L-functions at s =1.

1. INTRODUCTION

The

_following

_problems,

which arise in

_quite

different

_contexts,

have led

us to very similar

questions.

Problem A. This is a

problem

in

Diophantine

_geometry

and it was

suggested by

Cellini

_[2]

for the

_study

of the dimension of a commutative descent

_algebra

_{of the group}

_algebra

over

Q

of the

Weyl

_{groups of}

_type

Let _q be a

_positive

_integer

and for each a E Z consider the function

For

_{fixed q,}

how _{many among} the functions

_{f a~q}

are

_{linearly independent?}

It is clear that

_fa,q

_{depends only}

on the _congruence class of a modulo _q.

For a real number _a, let

[a]

denote its

_integer

_part,

denote its fractional

part

and

_{((a)) = {a} - 1/2.}

Then

whence is

_{1/g ’}

_{identity plus}

a function which is

periodic

modulo

q. Let also

~a~9(yt) _

((an/q)).

We shall prove the

following

Theorem 1. The dimension

_of

the vector _space

_{generated by}

the

_functions

ga,q is

equal

to

_{[(q -}

_3)/2]

-t-

_d(q)

and the dimension

_of

the _{vector space}

generated

by

the

_functions

_fa,q

is

_equal

to

_{[(q - 1)/2]}

+

_d(q),

where

_d(q)

is the number

_of

divisors

_of

q.

To prove this we have to

_study

the rank of the matrix

and this will be done in

_Corollary

_2,

where the rank will be

_given

in terms

of the number of

_{non-vanishing}

_expressions

of the form

where X

is a Dirichlet character with modulus a divisor d of _q. The method uses a

decomposition

of the _space of

functions If

f : Z/qZ -

C}

into

an

_orthogonal

sum of

_subspaces

defined in terms of the

_primitive

Dirichlet characters modulo divisors of _q.

_Although

this

_{decomposition}

follows rather

easily

from the classical

_theory

of Dirichlet

_characters,

we do not have

explicit

references for this

result,

so we include its

_complete

proof.

It is rather

_surprising

that

_Corollary

2 turns out to be

_equivalent

to

the

_{non-vanishing}

of the first Bernoulli numbers relative to odd characters

x, which in turn is

_equivalent,

via the functional

_equation,

to the

non-vanishing

of the

_{corresponding}

Dirichlet functions

_{L(s, X)}

at s =1.

Problem B. Let q be a

positive

integer

and let

f :

Z - C be odd and

is

_convergent

_everywhere

_(see

Lemma 7

_below).

Let

_8h(d)

denote the

Rie-mann sum

where d &#x3E; 1 is an

_integer

and h is

_{given by}

(4).

Our

problem

is the

following:

is it

_possible

to have

_6h(d)

= 0 for _every d &#x3E; 1 and the

corresponding

f (n)

not all zero? The answer is

_negative

(see

Theorem

2)

and is

_equivalent

to

proving

that the rank of the matrix

_{P(ij/q) (1}

_{i, j}

_q),

where

is

_{~(q -}

_1)/2~.

In the

_proof

a vital

_step

is

_{provided by}

the well known

_identity

(see

for instance

_[3]

and

_[4])

which can be deduced from

L(1, x) ~

0 and the convergence of the series

for all

_{non-principal}

Dirichlet

_{characters X}

modulo q.

2. SOLUTION TO PROBLEM A

Let _q &#x3E; 1 be an

integer.

We shall denote

by 0

a character modulo a

divisor of _q. If the modulus

_m(,O)

_{of o}

is

_q/d

for some divisor d of

q, 1b

will

also be denoted

_by

_0(d).

We shall denote

_by

or

by

_{f d,~}

if we want to

make

_explicit

the

_modulus,

the function defined on

(Z/qZ)

into the

_complex

numbers such that

We abbreviate

_Z/qZ

_by

_(q)

and set V =

(q)

_-

C}.

The scalar

product

of two functions

_f,g

E V is

_{given by}

Lemma 1. There are

_exactly

_q

_{functions of}

the

_type

these

functions

are

pairwise orthogonal

and

_generate

V as a vector _space. Furthermore

Proof.

The first assertion follows from the fact that there are

Ø(q/d)

char-acters modulo

_q/d

and be characters as above.

We have

Set d =

hdi,

_e =

hel

with

(el, dl)

=

1,

hence

[d,e]

=

hdlel.

Letting

X =

y[d, e]

one sees that the last sum

_equals

Now

_hdlel

=

del

divides _q, whence

I

(q/d)

and,

similarly,

di

q/e.

Since

1/J and 6

have modulus

_q/d

and

_q/e

_{respectively,}

we

_get

that = 0

unless ei =

d1

=

_1,

that is d = _e. In this case

and the lemma is

_proved.

0

Let X

be a

primitive

character.

If X induces 0

we

_{write X ~}

1 0.

Let

_Vx

be

the vector _space

_generated

_by

the such

Lemma 2. V is the direct sum

of

the

subspaces

Vx,

where _X runs over all

primitive

characters whose modulus is a divisor

of

_q.

Proof.

Clear. F-1

The scalar

_product

on V induces a

_pairing

as follows:

To

_study

the

_properties

of this

_pairing

a lemma will be useful.

If 9

is a

character with modulus

_q/d

and h is a _{divisor of d} we denote

by 9h

the

By

definition we have

Lemma 3. The

_{following equality}

holds:

Then the

_right-hand

term

_equals

The modulus of

_1Ph/v

is

_qh/dv.

Hence if

_{(hlv, r/v)}

&#x3E; 1 we have

0. On the other hand

_{(r/v) ~}

_{I (h/v),}

whence our sum contains at most

one

non-vanishing

_term,

corresponding

to v = _r,

namely

(xl).

Now

(qh1/d, Xl) = (q/d, xi), hence ’Øhl (Xl) =

In conclusion

_{(Xl) =}

- # ...1" , , I ,

-Lemma 4. For a

character V) of

modulus

q/d

the

following formula

holds:

Proof.

Let d =

di h,

y =

Y1h,

(dl,

yl)

= 1. Since

f1/J(xy)

= 0 if d

I

_{xy, we} assume that d

xy,

_or,

equivalently,

d, I x

and x =

d1x’.

In this case

Using

Lemma 3 with x = x’ =

x/dl

_we obtain

Let

and therefore

whenever

_dit

_x.

If

_d1

_X

x then all terms vanish and the formula holds

again.

0

Proof.

The first statement follows

_easily

from Lemma 1 and Lemma 4.

Suppose

now

_{that ~ =}

1Pp..

If p

fy

we have

Ifo,

= 0 and the _same is

true for the

_{righthandside}

of

_(10).

_{If p}

_y

one has

Applying

Lemma 3 with h =

d/ J.1

_we have

and,

setting x

=

y/p,

Corollary

1. The

_function

_{rF :}

V - V

_given.

_by

maps

_Yx

into

_Vx

and vice-versa.

where X

runs over

_primitive

characters with modulus

_dividing

_q and

_Fx

E

vx,

and set

Let ~

be a character induced

by

_X.

From Lemma

5,

putting

m(~)

=

q/r,

_we

get

Let X

=

x~a&#x3E;,

that is

qla

=

~n(X).

Then

ria.

If 9 =

then d

I

_a.

Moreover 9 [ g

implies

rid

whence

_rid

_I

a. Since also s

I

_a, is

equivalent

to

_(q/d)(r/v)

=

q/s,

that is rs = dv.

If X

is a non-real

(resp.

real)

character,

the matrix associated to the

restriction of the function

rF

to

_{Vx e Vg}

_(resp.

_Vx)

is of

_type

(resp.

of

_type

_{(A) )}

where A is the matrix of a linear function

r

on a vector

space with basis such that

where 1b

is the

_unique

character of modulus

_q/d

induced

_{by X}

and

_(3d

=

Q1jJo The conditions on d and v can also be written as v ~

(r,

s)

and d =

(rs/v) ~ a.

Moreover ,

otherwise. Hence

₍₁₁₎

becomes

We note that

Further,

if v satisfies

(12) then v

I

(r, s)

since

r, s

I

_q; so the inner sum

Lemma 6.

We consider now the

special

case

F(x) _ ((x/q)).

Observe that in this case

hence

_rF

can be

represented by

the matrix M defined in

(3).

Proposition

1. The rank

_of

_TF

is maximal on

_V.

+

_Vg

2uhen

_{L(X) ~}

0 and is zero when

L(X)

= 0.

Let o

be induced

_{by X}

=

X~a~

and =

q/d.

We have

But d

_I

a and

X(c)

= 0 if

(c, q/a)

_&#x3E; 1.

Hence,

since I

(a/d),

one _may assume c

I (a/d)

and

The inner sum is

equal

to

((a/qz)),

whence

It follows that if

_L(X)

= 0 then the linear function

T

is

identically

zero.

Assume

_{L(~) 7~}

0. The action of r is

_{given by}

where

_À(a/d)

is

_{multiplicative.}

Let

_6r,s

be the coefficients of

_YS

in the last

_expression.

We have to prove

that 0. Let _p be a

_prime

divisor of _q. For an

_integer

m we set

- _1- . - . , I." ... - - .1. - _I

Then

Let =

_{By multiplicativity}

of the functions involved _one

gets

If a =

pla*,

whence the matrix

6r,s

is of

_type

and

(see

for intance

_[1,

_Prop.

_2.14,

p.

202]

for the last

formula).

By

induction

on the number of

_prime

factors of a we obtain

for suitable

_integer

_exponents

_ep. Hence it suffices to _prove that

for any p. Consider _ep,,. We have = 0 if

min(p

+

_{a, K)}

&#x3E; a. Otherwise

Observe that =

_1,

~(p) _

À(p2) = ...

=

1 - ~(p).

Moreover

X(p) _

and the matrix becomes

and has non-zero determinant.

Suppose

now a = _. In this _{case we} have

- c# ""- _ - --"’- ... - ___ _ _

Regarding

note that a

_{non-vanishing}

factor of all _terms,

hence it _may be

_neglected.

Also we _may

neglect

X(pP+6)

=

by

first

_dividing

the

_p-th

row

_by

X(pP)

and after

_dividing

the cr-th column

_by

Setting 77

=

XA/0

_{we must} then

compute

that

is

, - .

-Subtracting

the second last column from the last one it becomes clear that

whence

On the other hand

Corollary

2. The rank

_{of rF is}

_{[(q - 1)/2]}

+

_d(q),

where

_d(n)

is the

num-ber

_of

divisors

_of

n.

Proof.

L(X)

= is in fact the first Bernoulli number associated with

X.

It is well known that

_Bl,x

= 0 for all

non-principal

_even

_characters,

whereas

5~

0

_{if x}

is odd _{or x is} the

_principal

character

_(see

for instance

_[5,

_p.

31]

for

_details).

_{By Proposition}

1 the rank of

_rF

is the sum of dimension

of

_vX

_{for X}

either odd or

principal.

The

principal

character X

= 1 induces

exactly

one character for each modulus

_{dividing q,}

hence

_{dim VI}

=

d(q).

Moreover,

a

_{character o}

is induced

_by

an odd

_primitive

_{character X}

_(or

is

odd

_equals

the number of odd characters modulo a divisor of _q. It is well

known

_that,

for

_{d ~}

_{1, 2,}

there are

_exactly

~(d)/2

odd

_characters,

while for

d

_{=1, 2}

there is

_only

one character

(the

principal

one).

Hence we have

and the

_corollary

follows. D

We

_finally

_go back to the

_proof

of Theorem 1.

Proof of

Theorem 1. Let _{jj :} V -~ V be the linear function defined

_by

The q

x _q matrix associated

_{to p}

is

Then the dimension of the vector space

spanned by

the functions ga,q is

equal

to the rank of the matrix

_BM,

where M has rank

_{~(q -1)/2~}

+

_d(q)

by

Corollary

2. Sincle

_clearly

B has rank

_{q -1,}

the rank of BM is

_equal

either to

_{rank(M) -}

1 or to

_rank(M)

_depending

on whether the

_image

of

rF

intersects or not the kernel of the linear function _p.

_Taking

for

_f

the function which is 1 at the zero class and zero

elsewhere,

we

_get

that

whence

_rF

is a constant and is contained in the kernel of _/~. As to the second

_part,

note that the

_equality

implies

that the

_{identity belongs}

to the vector space

generated by

the

f a,q,

which therefore can be

_spanned

_by

the

_identity

_together

with the ga,q. Now

the

_identity

is

_clearly

_linearly

_independent

from all the _{functions ga,q,} since

3. SOLUTION TO PROBLEM B We

_begin

_{by proving}

the

_following

Lemma 7. Let

_{f :}

Z - C be odd and

_periodic

modulo q. Then the series

converges

everywhere

and we have the

_identity

where

and

_P(x)

is

_given

_by

_(6).

Proof.

Let us consider the

_equivalent

relations

It is _easy to see that

_f

is odd if and

_only

if such is _g. First _suppose that

_f

is odd: then we have

The

_implication

in the other _{way is}

_{proved similarly.}

Since

_f

is odd and

_periodic

modulo _q so

_{is g}

and this

_implies

that

and

If we substitute

₍₁₄₎

for

_{f (n)}

and

_exchange

the sums we

_get

Taking

into account

₍₁₅₎

and

₍₁₆₎

we can write

(17)

in the form

Since we see that

(13)

follows from

(18)

when N -~ 00.

~

0

We now _prove that the coefficients

f (n)/n

of the function

h(x)

can be

recovered from the

_{corresponding}

Riemann sums

6h (d)

_{given by}

(5)

of the

introduction.

Theorem 2. Let

_h(x)

= _cos 27rnx _as in Lemma 7. Then _we have

Proof. By

(13)

of Lemma 7 we can write

Since . for _{every x} E R we have

It is _easy to see that

From

₍₂₁₎

and

₍₂₂₎

follows the

_identity

so that

(20)

becomes

From

₍₂₃₎

we obtain

Now recall that

,-.

in

₍₂₄₎

_by

the first one of the two

_reciprocal

relations

₍₁₄₎

we

_get

Theorem 2

_implies

the

_following

Corollary

3. For _every

_integers

_{q &#x3E;}

3 we have

Proof.

We shall show that if we _suppose that

(25)

is false for an

_integer

q &#x3E;

3 we reach a contradiction.

If

₍₂₅₎

is false then there exist values

_g(m)

not all zero such that

Let us now consider the odd continuation with

_{period q}

of the

g(m)

which are solutions of the

_system

(26),

i.e. we

require

that g :

Z - C is

_periodic

modulo q

and

_{g(q -}

_{m) = -g(m)}

V m E Z.

Let

_{f (n),}

_{n E Z,}

be defined

by

the first relation in

_(14),

that is

_{f (n) _}

~2013i

As

_already

noted

_{f (n)}

is also odd and

periodic

modulo _q: moreover since

the

_g(m)

are not all zero, the same is true for the

_{f (n).}

Let us now

_put

-cos 2Jrnz as in lemma 1.

_Equality

₍₂₃₎

can be written in

the form

because we have

since

_{both 9}

and P are odd.

From

₍₂₆₎

and

₍₂₇₎

it follows

_{6h (d)}

= 0 V d &#x3E; 1 which

_{implies, by}

₍₁₉₎

of Theorem

_2,

_{f (d)}

= 0 V d &#x3E; 1. But this is a

_{contradiction,}

since

f

is not

identically

zero. This _proves the

corollary.

D

REFERENCES

[1] W. A. ADKINS &#x26; S. H. WEINTRAUB - Algebra, An _Approach via Module Theory, Springer Verlag (1992).

[2] P. CELLINI - A general commutative descent _algebra, Journal _{of Algebra} 175 _(1995), 990-1014.

[3] H. DAVENPORT - On some infinite series _involving arithmetical functions, Quarterly Journal

of Mathematics 8 _(1937), 8-13.

[4] S.L. SEGAL - On an _identity between infinte series and arithmetic functions, Acta Arith-metica XXVIII _(1976), 345-348.

[5] L.C. WASHINGTON - Introduction to _Cyclotomic Fields, Springer-Verlag (1982).

Paolo CODECK

Dipartimento di Matematica

via _Machiavelli, 35 44100 FERRARA - ITALY E-mail : cododns. unif e. it

Roberto DVORNICICH Dipartimento di Matematica via Buonarroti, 2 56127 PISA - ITALY E-mail : _{dvornic0gauss.dm.unipi.it} Umberto ZANNIER Ist.Univ.Arch.Venezia D.S.T.R. S.Croce, 191 30135 VENEZIA - ITALY E-mail : zannier0brezza.iuav.unive.it