Unified Treatment of Explicit and Trace Formulas via Poisson–Newton Formula

Abstract

We prove that a Poisson–Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane of finite order. These formulas simultaneously generalize the classical Poisson formula and Newton formulas for Newton sums. Classical Poisson formulas in Fourier analysis, explicit formulas in number theory and Selberg trace formulas in Riemannian geometry appear as special cases of our general Poisson–Newton formula.

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Correspondence to Vicente Muñoz.

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We dedicate this article to Daniel Barsky and Pierre Cartier for their interest and constant support.

Communicated by A. Connes

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Muñoz, V., Pérez Marco, R. Unified Treatment of Explicit and Trace Formulas via Poisson–Newton Formula. Commun. Math. Phys. 336, 1201–1230 (2015). https://doi.org/10.1007/s00220-015-2312-1

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Keywords

  • Functional Equation
  • Meromorphic Function
  • Zeta Function
  • Dirichlet Series
  • Trace Formula