The second and fourth moments of theta functions at their central point

Stéphane R. Louboutin; Marc Munsch

doi:10.1016/j.jnt.2012.09.016

The second and fourth moments of theta functions at their central point

Stéphane R. Louboutin^*, Marc Munsch

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

Let χ range over the (p - 1)/2 even Dirichlet characters mod p ≥ 3, a prime. Let θ(x, χ) be the associated theta series. It is known that the square mean value of θ(1, χ) is asymptotic to p3/2/42 as p goes to infinity. We prove that the fourth mean value of θ(1, χ) is asymptotic to 3/16πp²logp as p goes to infinity. We give similar results for mean values of odd Dirichlet characters mod p.

Original language	English
Pages (from-to)	1186-1193
Number of pages	8
Journal	Journal of Number Theory
Volume	133
Issue number	4
DOIs	https://doi.org/10.1016/j.jnt.2012.09.016
Publication status	Published - 1 Apr 2013
Externally published	Yes

Keywords

Dirichlet characters
Mean values
Theta functions

ASJC Scopus subject areas

Algebra and Number Theory

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AU - Munsch, Marc

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AB - Let χ range over the (p - 1)/2 even Dirichlet characters mod p ≥ 3, a prime. Let θ(x, χ) be the associated theta series. It is known that the square mean value of θ(1, χ) is asymptotic to p3/2/42 as p goes to infinity. We prove that the fourth mean value of θ(1, χ) is asymptotic to 3/16πp2logp as p goes to infinity. We give similar results for mean values of odd Dirichlet characters mod p.

KW - Dirichlet characters

KW - Mean values

KW - Theta functions

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JF - Journal of Number Theory

IS - 4

ER -

The second and fourth moments of theta functions at their central point

Abstract

Keywords

ASJC Scopus subject areas

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