A conjectural extension of Hecke’s converse theorem

Sandro Bettin, Jonathan Bober, Andrew R. Booker, Brian Conrey, Min Lee, Giuseppe Molteni, Thomas Oliver, David J. Platt, Raphael S. Steiner

Type: Article

Publication Date: 2017-11-10

Citations: 9

DOI: https://doi.org/10.1007/s11139-017-9953-y

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Abstract

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.

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The Ramanujan Journal - View - PDF
arXiv (Cornell University) - View - PDF
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Works Cited by This (19)

Action	Title	Year	Authors
+	Multiplicative Number Theory	1980	H. Davenport
+	Hecke's Theory of Modular Forms and Dirichlet Series	2007	Bruce C. Berndt Marvin I. Knopp
+	Automorphic forms and representations	1997	Daniel Bump
+	An extension of Hecke's converse theorem	1995	J. Brian Conrey David W. Farmer
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+	�ber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen	1967	Andr� Weil
+	Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung	1936	E. Hecke
+	The divisors of <i>p</i> ‐1	1973	R. R. Hall
+	An Arithmetic-Geometric Method in the Study of the Subgroups of the Modular Group	1991	Ravi S. Kulkarni
+ PDF Chat	A practical method for enumerating cosets of a finite abstract group	1936	John A. Todd H. S. M. Coxeter