Mean-periodicity and zeta functions

TL;DR

This paper explores the deep connection between zeta functions of arithmetic schemes and mean-periodic functions, extending classical correspondences and providing new insights into their analytic properties.

Contribution

It establishes a novel link between the meromorphic continuation of zeta functions and mean-periodicity, extending the Hecke--Weil correspondence to a broader class of functions.

Findings

Meromorphic continuation linked to mean-periodicity

Extension of Hecke--Weil correspondence

Detailed analysis of elliptic curves over number fields

Abstract

This paper establishes new bridges between number theory and modern harmonic analysis, namely between the class of complex functions, which contains zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of mean-periodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and functional equation of the zeta function of arithmetic scheme with its expected analytic shape is shown to correspond to mean-periodicity of a certain explicitly defined function associated to the zeta function. This correspondence can be viewed as an extension of the Hecke--Weil correspondence. The case of elliptic curves over number fields and their regular models is treated in more details, and many other examples are included as well.