Analytic $L$-functions: Definitions, Theorems, and Connections

TL;DR

This paper proposes a set of axioms to unify the analytic and automorphic perspectives of $L$-functions, establishing a framework that bridges Selberg's axioms and Langlands' automorphic constructions.

Contribution

It introduces a new collection of axioms that connect the general analytic and algebraic descriptions of $L$-functions, and proves related theorems and conjectures.

Findings

The axioms successfully bridge the gap between Selberg's and Langlands' descriptions.

Theorems about $L$-functions satisfying these axioms are established.

Conjectures naturally arise from the proposed axiomatic framework.

Abstract

$L$-functions can be viewed axiomatically, such as in the formulation due to Selberg, or they can be seen as arising from cuspidal automorphic representations of $\textrm{GL}(n)$, as first described by Langlands. Conjecturally these two descriptions of $L$-functions are the same, but it is not even clear that these are describing the same set of objects. We propose a collection of axioms that bridges the gap between the very general analytic axioms due to Selberg and the very particular and algebraic construction due to Langlands. Along the way we prove theorems about $L$-functions that satisfy our axioms and state conjectures that arise naturally from our axioms.