A conjectural extension of Hecke’s converse theorem

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

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

6 Citations (Scopus)
311 Downloads (Pure)


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.

Original languageEnglish
Pages (from-to)659-684
Number of pages26
JournalRamanujan Journal
Issue number3
Early online date10 Nov 2017
Publication statusPublished - Dec 2018


  • Modular forms
  • Converse theorems
  • Ramanujan sums


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