On twisted period functions and Moments of a weighted mean square of Dirichlet L-functions on the critical line

TL;DR

This paper generalizes the theory of period functions and Eisenstein series to Dirichlet L-functions with arbitrary primitive characters, deriving new formulas for moments of their mean squares on the critical line.

Contribution

It extends existing frameworks to a broader class of L-functions and introduces novel closed-form expressions for their moments, along with an asymptotic formula based on Euler--Maclaurin summation.

Findings

Derived closed-form expressions for moments of weighted mean squares of Dirichlet L-functions.

Established positivity conditions valid for all primitive characters.

Developed an asymptotic formula using shifted Euler--Maclaurin summation.

Abstract

We extend to Dirichlet L-functions associated with arbitrary primitive characters a range of objects and properties -- including Eisenstein series and period functions -- that were originally introduced and studied by Lewis and Zagier (2001), and later by Bettin and Conrey (2013) in the case of the Riemann zeta function, and more recently by Lewis and Zagier (2019) for odd real characters. These tools yield closed-form expressions for the moments of a measure defined via a weighted mean square of the L-function. These moments not only provide a complete characterization of the modulus of the L-function on the critical line, but also imply an infinite number of non-trivial positivity conditions valid for all primitive characters, real or not. The methods also involve a general form of an asymptotic formula based on the shifted Euler--Maclaurin summation formula, which may be of independent interest.