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.
A conjectural extension of Hecke’s converse theorem / S. Bettin, J.W.B.. - In: RAMANUJAN JOURNAL. - ISSN 1382-4090. - 47:3(2018 Dec 01), pp. 659-684. [10.1007/s11139-017-9953-y]
A conjectural extension of Hecke’s converse theorem
G. Molteni;
2018
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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