On the fourth moment of theta functions at their central

point

Amadou Diogo BARRY and St´ ephane R. LOUBOUTIN

Abstract. Let χ be a Dirichlet character of prime conductor p ≥ 3. Set A =

(χ(1) − χ(−1))/2 ∈ {0, 1} and let θ(x, χ) =

P

n≥1

nAχ(n) exp(−πn2x/p)

(x

0) be its associated theta series whose functional equation is used to obtain

the analytic continuation and functional equation of the L-series L(s, χ) =

P

n≥1

χ(n)n−s.

These functional equations depend on some root numbers

W (χ), complex numbers of absolute values equal to one. In particular, if

θ(1, χ) = 0, then numerical approximations to W (χ) = θ(1, χ)/θ(1, χ) can

be eﬃciently computed, which leads to a fast algorithm for computing class

numbers and relative class numbers of real or imaginary abelian number ﬁelds

(see the bibliography). According to numerical computations, it is reasonable

to conjecture that θ(1, χ) = 0 for any primitive Dirichlet character χ. One way

to prove that this conjecture at least holds true for inﬁnitely many primitive

characters is to study the moments

P

χ

|θ(1,

χ)|2k

for k ∈ Z≥1, where χ ranges

over all the even or odd primitive Dirichlet characters of conductor p. This

paper is devoted to proving a lower bound on these moments for 2k = 4.

1. Introduction

We restrict ourselves to even characters of prime conductors. Let p 3 be a

prime. Let Xp

+

be the set of the (p − 3)/2 primitive even Dirichlet characters of

conductor p. For χ ∈ Xp +, set

θ(x, χ) =

n≥1

χ(n)e−πn2x/p

(x 0).

The analytic continuation and the functional equation satisﬁed by the Dirichlet

L-series L(s, χ) =

∑

n≥1

χ(n)n−s,

(s) 0, stem from the functional equation

satisﬁed by the associated theta function (see [Dav, Chapter 9]):

(1) θ(x, χ) =

W (χ)

x1/2

θ(1/x, ¯), χ

where W (χ) = τ (χ)/

√

p (the Artin root number, a complex number of absolute

value equal to 1 ) with τ (χ) =

∑p

k=1

χ(k)e2πik/p (Gauss sum).

1991 Mathematics Subject Classiﬁcation. 2000 Mathematics Subject Classiﬁcation. Primary

11R42, 11N37. Secondary 11M06.

Key words and phrases. Dirichlet characters, Artin root numbers, Theta functions, Divisor

function.

1

Contemporary Mathematics

Volume 487, 2009

1

point

Amadou Diogo BARRY and St´ ephane R. LOUBOUTIN

Abstract. Let χ be a Dirichlet character of prime conductor p ≥ 3. Set A =

(χ(1) − χ(−1))/2 ∈ {0, 1} and let θ(x, χ) =

P

n≥1

nAχ(n) exp(−πn2x/p)

(x

0) be its associated theta series whose functional equation is used to obtain

the analytic continuation and functional equation of the L-series L(s, χ) =

P

n≥1

χ(n)n−s.

These functional equations depend on some root numbers

W (χ), complex numbers of absolute values equal to one. In particular, if

θ(1, χ) = 0, then numerical approximations to W (χ) = θ(1, χ)/θ(1, χ) can

be eﬃciently computed, which leads to a fast algorithm for computing class

numbers and relative class numbers of real or imaginary abelian number ﬁelds

(see the bibliography). According to numerical computations, it is reasonable

to conjecture that θ(1, χ) = 0 for any primitive Dirichlet character χ. One way

to prove that this conjecture at least holds true for inﬁnitely many primitive

characters is to study the moments

P

χ

|θ(1,

χ)|2k

for k ∈ Z≥1, where χ ranges

over all the even or odd primitive Dirichlet characters of conductor p. This

paper is devoted to proving a lower bound on these moments for 2k = 4.

1. Introduction

We restrict ourselves to even characters of prime conductors. Let p 3 be a

prime. Let Xp

+

be the set of the (p − 3)/2 primitive even Dirichlet characters of

conductor p. For χ ∈ Xp +, set

θ(x, χ) =

n≥1

χ(n)e−πn2x/p

(x 0).

The analytic continuation and the functional equation satisﬁed by the Dirichlet

L-series L(s, χ) =

∑

n≥1

χ(n)n−s,

(s) 0, stem from the functional equation

satisﬁed by the associated theta function (see [Dav, Chapter 9]):

(1) θ(x, χ) =

W (χ)

x1/2

θ(1/x, ¯), χ

where W (χ) = τ (χ)/

√

p (the Artin root number, a complex number of absolute

value equal to 1 ) with τ (χ) =

∑p

k=1

χ(k)e2πik/p (Gauss sum).

1991 Mathematics Subject Classiﬁcation. 2000 Mathematics Subject Classiﬁcation. Primary

11R42, 11N37. Secondary 11M06.

Key words and phrases. Dirichlet characters, Artin root numbers, Theta functions, Divisor

function.

1

Contemporary Mathematics

Volume 487, 2009

1