The Voronoi formula and double Dirichlet series

TL;DR

This paper establishes a generalized Voronoi formula for a broad class of $L$-function coefficients using double Dirichlet series, expanding applicability beyond automorphic forms.

Contribution

It introduces a new proof of the Voronoi formula that relies on functional equations and double Dirichlet series, not requiring automorphy.

Findings

Proves a Voronoi formula for various $L$-functions.

Constructs a double Dirichlet series as a key tool.

Extends the applicability of the Voronoi formula to non-automorphic cases.

Abstract

We prove a Voronoi formula for coefficients of a large class of $L$-functions including Maass cusp forms, Rankin-Selberg convolutions, and certain isobaric sums. Our proof is based on the functional equations of $L$-functions twisted by Dirichlet characters and does not directly depend on automorphy. Hence it has wider application than previous proofs. The key ingredient is the construction of a double Dirichlet series.