Sandro Bettin

THE

We show that the θ = ∞ conjecture implies the Riemann hypothesis.

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We show that the θ = ∞ conjecture implies the Riemann hypothesis.

CONJECTURE IMPLIES THE RIEMANN HYPOTHESIS

We compute the first moment of twisted Hecke

The

L

\theta=\infty

-functions of prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.

conjecture implies the Riemann hypothesis

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.