2004/03/02 by Stephen D. Miller, Miller, Stephen D., Wilfried Schmid +1
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Number Theory (math.NT) #advanced mathematical theories #math.CA #math.FA #math.NT
paper · pdf · doi:10.48550/arxiv.math/0403030
72 pages
arxiv created 2004/03/02 · openalex publication_date 2004/03/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is second in a series of three papers; the first of which is "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187), and the third of which is "Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)". The first paper is primarily an expository paper, while the third proves a Voronoi-style summation formula for the coefficients of a cusp form on GL(3,Z)\GL(3,R). This 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)xGL(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 Riemann's 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.