A conjectural extension of Hecke's converse theorem

We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture...

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Bibliographic Details
Published in:arXiv.org 2017-09
Main Authors: Bettin, Sandro, Bober, Jonathan W, Booker, Andrew R, Conrey, Brian, Lee, Min, Molteni, Giuseppe, Oliver, Thomas, Platt, David J, Steiner, Raphael S
Format: Article
Language:English
Subjects:
Dirichlet problem
Hypotheses
Theorems
Online Access:Get full text
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Summary:We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.
ISSN:2331-8422
DOI:10.48550/arxiv.1704.02570