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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Analytic $ L$-functions: Definitions, theorems, and connections


Authors: David W. Farmer, Ameya Pitale, Nathan C. Ryan and Ralf Schmidt
Journal: Bull. Amer. Math. Soc. 56 (2019), 261-280
MSC (2010): Primary 11M06, 11M41, 11F66, 11F03, 11F70
DOI: https://doi.org/10.1090/bull/1646
Published electronically: September 21, 2018
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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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Additional Information

David W. Farmer
Affiliation: American Institute of Mathematics, 600 East Brokaw Road, San Jose, California 95112-1006
Email: farmer@aimath.org

Ameya Pitale
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019-3103
Email: apitale@math.ou.edu

Nathan C. Ryan
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837
Email: nathan.ryan@bucknell.edu

Ralf Schmidt
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019-3103
Email: rschmidt@math.ou.edu

DOI: https://doi.org/10.1090/bull/1646
Received by editor(s): December 9, 2017
Published electronically: September 21, 2018
Article copyright: © Copyright 2018 American Mathematical Society