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Thursday, October 29, 2015

**Abstract:** We present a proof of Voronoi formula for coefficients of a large class of L-functions, in the style of the classical converse theorem of Weil. Our formula applies to full-level 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 any previous proofs. The key ingredient is the construction of a double Dirichlet series associated with these coefficients and the structure of nonprimitive Gauss/Ramanujan sums. This is joint work with Eren Mehmet Kıral.