Distributions and analytic continuation of Dirichlet series

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)×GL(1) 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 Riemannʹs zeta function, degree 1 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.