Distributions and analytic continuation of Dirichlet series |
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Authors: | Stephen D Miller Wilfried Schmid |
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Affiliation: | a Department of Mathematics, Hill Center-Busch Campus, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA b Department of Mathematics, Harvard University, Cambridge, MA 02138, USA |
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Abstract: | 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 . 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 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. |
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Keywords: | Distributions L-functions Analytic Continuation Mellin tranform Dirichlet series Functional equation Riemann zeta function |
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