Distributions and analytic continuation of Dirichlet series

摘要

This is the second in a series of three papers; the other two are "Summation Formulas, from Poisson and Voronoi to the Present" [Progr. Math. 220 (2004) 419-440] and "Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)" (preprint). The first paper is primarily expository, while the third proves a Voronoi-style summation formula for the coefficients of a cusp form on GL(3, Z)GL(3, R). The present paper contains the distributional machinery used in the third paper for rigorously deriving the summation formula, and also for the proof of the GL(3) x GL(l) converse theorem given in the third paper. The primary concept studied is a notion of the order of vanishing of a distribution along a closed submanifold. Applications are given to the analytic continuation of Riemann's zeta function, degree I and degree 2 L-functions, the converse theorem for GL(2), and a characterization of the classical Mellin transform/inversion relations on functions with specified singularities. (C) 2004 Published by Elsevier Inc.