Poisson-Newton formulas and Dirichlet series

TL;DR

This paper introduces a general Poisson-Newton formula linked to Dirichlet series, unifying various classical formulas across Fourier analysis, number theory, and geometry, and extends to meromorphic functions.

Contribution

It generalizes classical Poisson and Newton formulas, connecting them through a broad framework applicable to Dirichlet series and meromorphic functions.

Findings

Unified framework for classical formulas in analysis and number theory

General Poisson-Newton formulas for meromorphic functions of finite order

Special cases include Euler-Maclaurin, Abel-Plana, and Selberg trace formulas

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. These formulas simultaneously generalize the classical Poisson formula and Newton formulas for Newton sums. Classical Poisson formulas in Fourier analysis, classical summation formulas as Euler-McLaurin or Abel-Plana formulas, explicit formulas in number theory and Selberg trace formulas in Riemannian geometry appear as special cases of our general Poisson-Newton formula. We also associate to finite order meromorphic functions general Poisson-Newton formulas that yield many classical integral formulas.