# Dirichlet twists of $\operatorname{GL}_n$-automorphic $L$-functions and hyper-Kloosterman Dirichlet series

## TL;DR

This paper develops new summation formulas for twisted automorphic $L$-functions and hyper-Kloosterman series, enabling analytic continuation and deeper understanding of their mean values at various points.

## Contribution

It introduces novel Voronoi summation formulae for prime-power moduli and expresses twisted sums in terms of their own averages, advancing the analysis of automorphic $L$-functions.

## Key findings

- Derived new Voronoi summation formulas for prime-power moduli.
- Established analytic continuation for hyper-Kloosterman Dirichlet series.
- Expressed twisted sums in terms of their own averages.

## Abstract

We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series associated to $\operatorname{GL}_n$-automorphic $L$-functions.