Breaking the $\frac{1}{2}$-barrier for the twisted second moment of Dirichlet $L$-functions

TL;DR

This paper advances the understanding of the second moment of Dirichlet L-functions by breaking the 1/2 barrier for twisted moments, providing new asymptotic formulas and bounds for moments of these functions.

Contribution

It introduces a novel method to surpass the 1/2 barrier in the twisted second moment of Dirichlet L-functions, with specific bounds on polynomial length and applications to higher moments.

Findings

Achieved asymptotic formula for twisted second moment with polynomial length less than q^{51/101}

Provided upper bounds for the third moment of Dirichlet L-functions

Extended results for specialized coefficients of Dirichlet polynomials

Abstract

We study the second moment of Dirichlet $L$-functions to a large prime modulus $q$ twisted by the square of an arbitrary Dirichlet polynomial. We break the $\frac{1}{2}$-barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than $q^{51/101} = q^{1/2 +1/202}$. As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet $L$-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.