Converse theorems: from the Riemann zeta function to the Selberg class

TL;DR

This paper reviews classical and modern converse theorems for L-functions, including developments within the Selberg class, highlighting their theoretical significance and recent advances in understanding their properties.

Contribution

It provides an expanded overview of converse theorems from classical cases to the Selberg class, integrating historical context and recent results.

Findings

Classical converse theorems are revisited and extended.

Recent developments in the Selberg class are discussed.

New results on converse theorems within the Selberg framework are presented.

Abstract

This is an expanded version of the author's lecture at the XX Congresso U.M.I., held in Siena in September 2015. After a brief review of L-functions, we turn to the classical converse theorems of H.Hamburger, E.Hecke and A.Weil, and to some later developments. Finally we present several results on converse theorems in the framework of the Selberg class of L-functions.