On the twisted quadratic moment for Dirichlet L-functions

TL;DR

This paper derives explicit and asymptotic formulas for a twisted quadratic moment involving Dirichlet L-functions, specifically summing over non-trivial characters with conductors greater than a given integer.

Contribution

It provides new explicit and asymptotic formulas for twisted quadratic moments of Dirichlet L-functions, advancing understanding of their behavior.

Findings

Explicit formula for the sum involving Dirichlet characters and L-values.

Asymptotic formula for the sum as the conductor grows.

Enhanced understanding of the distribution of L-values in character families.

Abstract

Given $c,$ a positive integer, we give an explicit formula and an asymptotic formula for \[ \sum\chi(c)|L(1,\,\chi)|^{2}, \] where $\chi$ is the non-trivial Dirichlet character mod $f$ with $f>c.$