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Analytic \(L\)-functions: definitions, theorems, and connections. (English) Zbl 1453.11107

Summary: \( L\)-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.

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M41 Other Dirichlet series and zeta functions
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F03 Modular and automorphic functions
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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