#### Vol. 10, No. 10, 2016

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The Voronoi formula and double Dirichlet series

### Eren Mehmet Kıral and Fan Zhou

Vol. 10 (2016), No. 10, 2267–2286
##### Abstract

We prove a Voronoi formula for coefficients of a large class of $L$-functions including Maass cusp forms, Rankin–Selberg convolutions, and certain noncuspidal forms. 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.

##### Keywords
Voronoi formula, automorphic form, Maass form, multiple Dirichlet series, Gauss sum, Kloosterman sum, Rankin–Selberg $L$-function
##### Mathematical Subject Classification 2010
Primary: 11F30
Secondary: 11F68, 11L05
##### Milestones
Received: 8 May 2016
Revised: 19 July 2016
Accepted: 23 September 2016
Published: 9 December 2016
##### Authors
 Eren Mehmet Kıral Department of Mathematics Texas A&M University College Station, TX 77843 United States Fan Zhou Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, OH 43210 United States