Convolution formula for the sums of generalized Dirichlet L-functions

TL;DR

This paper develops a spectral decomposition for sums of generalized Dirichlet L-functions using the Kuznetsov trace formula, leading to explicit formulas and asymptotic expansions connecting prime geodesics and L-function moments.

Contribution

It introduces a novel spectral decomposition approach for generalized Dirichlet L-functions and derives explicit formulas linking geometric and analytic number theory.

Findings

Derived an explicit formula relating prime geodesic norms to L-function moments

Provided an asymptotic expansion for average central values of generalized Dirichlet L-functions

Applied the Kuznetsov trace formula to spectral analysis of L-functions

Abstract

Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet $L$-functions. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square $L$-functions and an asymptotic expansion for the average of central values of generalized Dirichlet $L$-functions.