Dirichlet series from automorphic forms

This integral trick assumes greater significance when the function f is known to have strong decay properties both at 0 and at ∞, since then the Mellin transform is entire in s. One way to ensure such rapid decay is via eigenfunction properties in the context of automorphic forms. [2] • The archetype Mellin transform: zeta from theta • Abstracting to holomorphic modular forms • Variation: waveforms • Appendix: proofs of Poisson summation [1] The identity follows by changing variables, replacing y by ny. [2] Further, Converse Theorems, developed by Hecke, Weil, Jacquet-Langlands, Piatetski-Shapiro, and many others, assert very roughly that any Dirichlet series (perhaps with Euler product) with meromorphic continuation and functional equation comes from an automorphic form of some sort. This very naive claim (roughly Hecke’s form) is false outside the very simplest cases, as Weil already illustrated by his converse theorem, whose hypothesis needs the meromorphic continuation of a larger family of so-called twists of the original Dirichlet series. Consideration of much larger families of twists play a role in all modern converse theorems.