Analytic 𝐿-functions: Definitions, theorems, and connections

Farmer, David; Pitale, Ameya; Ryan, Nathan; Schmidt, Ralf

doi:10.1090/bull/1646

Analytic $L$-functions: Definitions, theorems, and connections
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by David W. Farmer, Ameya Pitale, Nathan C. Ryan and Ralf Schmidt PDF

Bull. Amer. Math. Soc. 56 (2019), 261-280 Request permission

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 $\operatorname {GL}(n)$, 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 representation-theoretic 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.

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