L-functions as distributions

TL;DR

This paper introduces an axiomatic framework for L-functions, extending the Selberg class, and demonstrates new applications such as the existence of infinitely many zeros of odd order for certain automorphic L-functions.

Contribution

It develops a distributional approach to L-functions, allowing for broader applications and new results like zeros of odd order in automorphic cases.

Findings

L-functions can be characterized by distributional identities.

Automorphic L-functions on GL_3 have infinitely many zeros of odd order.

The approach generalizes traditional conditions like Euler products and functional equations.

Abstract

We define an axiomatic class of L-functions extending the Selberg class. We show in particular that one can recast the traditional conditions of an Euler product, analytic continuation and functional equation in terms of distributional identities akin to Weil's explicit formula. The generality of our approach enables some new applications; for instance, we show that the L-function of any cuspidal automorphic representation of GL_3(A_Q) has infinitely many zeros of odd order.