The Selberg trace formula as a Dirichlet series

TL;DR

This paper investigates expressing the Selberg trace formula as a Dirichlet series, offering new interpretations of eigenvalue conjectures and formulas for sums of symmetric square L-functions related to Maass forms.

Contribution

It introduces a novel approach to represent the Selberg trace formula as a Dirichlet series, connecting eigenvalue conjectures and L-function sums.

Findings

Interpretation of the Selberg eigenvalue conjecture via quadratic twists

Derived a formula for arithmetically weighted sums of symmetric square L-functions

Enhanced understanding of the spectral theory of automorphic forms

Abstract

We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in terms of quadratic twists of certain Dirichlet series, and a formula for an arithmetically weighted sum of the complete symmetric square L-functions associated to cuspidal Maass newforms of squarefree level N>1.