Deux extensions de Th\'eor\`emes de Hamburger

TL;DR

This paper extends Hamburger's theorems on Dirichlet series with functional equations similar to the Riemann zeta function under weaker assumptions, using a correspondence with tempered distributions.

Contribution

It introduces two new extensions of Hamburger's theorems based on a novel link between meromorphic functions and tempered distributions with extended support conditions.

Findings

Extended theorems under weaker hypotheses

Established a correspondence between meromorphic functions and distributions

Broadened the class of functions satisfying the functional equation

Abstract

We propose two types of extensions to Hamburger's theorems on the Dirichlet series with functional equation like the one of the Riemann zeta function, under weaker hypotheses. This builds upon the dictionary betweeen the moderate meromorphic functions with functional equation and the tempered distributions with extended S-support condition. ----- Nous proposons deux types d'extensions aux th\'eor\`emes de Hamburger sur les s\'eries de Dirichlet avec \'equation fonctionnelle comme celle de la fonction z\^eta de Riemann, sous des hypoth\`eses plus faibles. Ceci repose sur le dictionnaire entre les fonctions m\'eromorphes mod\'er\'ees avec cette \'equation fonctionnelle et les distributions temp\'er\'ees avec la condition de support S-\'etendue.