Unified treatment of Explicit and Trace Formulas via Poisson-Newton formula

TL;DR

This paper introduces a unified Poisson-Newton formula that generalizes classical Fourier, number theory, and geometric trace formulas, applicable to broad classes of Dirichlet series with meromorphic extensions.

Contribution

It establishes a general Poisson-Newton formula linking explicit, Newton, and trace formulas within a single framework for Dirichlet series.

Findings

General Poisson-Newton formula encompasses classical formulas

Unifies Fourier, number theory, and geometric trace formulas

Applicable to Dirichlet series with meromorphic extensions of finite order

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.