Secondary terms in the asymptotics of moments of L-functions

TL;DR

This paper refines conjectural asymptotic formulas for moments of quadratic Dirichlet L-functions over function fields, providing insights into their analytic properties and potential extensions to number fields.

Contribution

It introduces a refined conjectural asymptotic formula for quadratic L-function moments, based on two conjectures that determine the generating function's analytic structure.

Findings

Proposes a refined asymptotic formula for moments over function fields.

Connects the formula to the analytic properties of a generating function.

Simplifies analysis by focusing on rational function fields.

Abstract

We propose a refined version of the existing conjectural asymptotic formula for the moments of the family of quadratic Dirichlet L-functions over rational function fields. Our prediction is motivated by two natural conjectures that provide sufficient information to determine the analytic properties (meromorphic continuation, location of poles, and the residue at each pole) of a certain generating function of moments of quadratic L-functions. The number field analogue of our asymptotic formula can be obtained by a similar procedure, the only difference being the contributions coming from the archimedean and even places, which require a separate analysis. To avoid this additional technical issue, we present, for simplicity, the asymptotic formula only in the rational function field setting. This has also the advantage of being much easier to test.