Effective Log-Free Zero Density Estimates for Automorphic L-Functions and the Sato-Tate Conjecture

Robert J. Lemke Oliver, Jesse Thorner

Research output: Contribution to journalArticlepeer-review

Abstract

Let K/Q be a number field. Let πand π′ be cuspidal automorphic representations of GLd(AK) and GLd(AK) We prove an unconditional and effective log-free zero density estimate for all automorphic L-functions L(s, π) and prove a similar estimate for Rankin-Selberg L-functions L(s, π× π′) when πor π′ satisfies the Ramanujan conjecture. As applications, we make effective Moreno's analog of Hoheisel's short interval prime number theorem and extend it to the context of the Sato-Tate conjecture; additionally, we bound the least prime in the Sato-Tate conjecture in analogy with Linnik's theorem on the least prime in an arithmetic progression. We also prove effective log-free density estimates for automorphic L-functions averaged over twists by Dirichlet characters, which allows us to prove an "average Hoheisel" result for GLdL-functions.

Original languageEnglish (US)
Pages (from-to)6988-7036
Number of pages49
JournalInternational Mathematics Research Notices
Volume2019
Issue number22
DOIs
StatePublished - Nov 18 2019
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics

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