The reciprocity law for the twisted second moment of Dirichlet L-functions

TL;DR

This paper provides a new proof of the reciprocity law for the twisted second moment of Dirichlet L-functions, using analysis of two-variable sums satisfying linear congruences, extending the known reciprocity results.

Contribution

It introduces a novel proof technique based on two-variable sums and extends the uniformity range of Conrey's reciprocity formula for Dirichlet L-functions.

Findings

New proof of the reciprocity law for twisted second moments

Reciprocity formula for sums satisfying linear congruences

Extended uniformity range of Conrey's reciprocity formula

Abstract

We produce a new proof of the reciprocity law for the twisted second moment of Dirichlet L-functions that was recently proved by Conrey. Our method is to analyze certain two-variable sums where the variables satisfy a linear congruence. We show that these sums satisfy an elegant reciprocity formula. In the case that the modulus is prime, these sums are closely related to the twisted second moment, and the reciprocity formula for these sums implies Conrey's reciprocity formula. We also extend the range of uniformity of Conrey's formula.