The mean square of the product of a Dirichlet $L$-function and a Dirichlet polynomial

TL;DR

This paper develops an asymptotic large sieve to analyze the mean square of the product of Dirichlet L-functions and polynomials across all characters with conductors up to Q, extending previous conjectures.

Contribution

It proves an analogue of a conjecture for Dirichlet characters with conductors up to Q using an advanced asymptotic large sieve technique.

Findings

Established an asymptotic formula for the mean square

Extended conjectures to a broader family of Dirichlet characters

Applied large sieve methods to L-function analysis

Abstract

We establish an analogue of a conjecture of Balasubramanian, Conrey, and Heath-Brown for the family of all Dirichlet characters with conductor up to $Q$. This forms another application of our work in developing an asymptotic large sieve.