A Bombieri-Vinogradov Theorem for Higher-Rank Groups

Yujiao Jiang, Guangshi Lü, Jesse Thorner, Zihao Wang

Research output: Contribution to journalArticlepeer-review

Abstract

We establish a result of Bombieri-Vinogradov type for the Dirichlet coefficients at prime ideals of the standard L-function associated to a self-dual cuspidal automorphic representation π of GLn over a number field F when π is not a quadratic twist of itself. Our result does not rely on any unproven progress towards the generalized Ramanujan conjecture or the nonexistence of Landau-Siegel zeros. In particular, when π is fixed and not equal to a quadratic twist of itself, we prove the first unconditional Siegel-type lower bound for the twisted L-values |L(1, π ⊗ χ)| in the χ-aspect, where χ is a primitive quadratic Hecke character over F. Our result improves the levels of distribution in other works that relied on these unproven hypotheses. As applications, when n = 2, 3, 4, we prove a GLn analogue of the Titchmarsh divisor problem and a nontrivial bound for a certain GLn × GL2 shifted convolution sum.

Original languageEnglish (US)
Pages (from-to)482-535
Number of pages54
JournalInternational Mathematics Research Notices
Volume2023
Issue number1
DOIs
StatePublished - Jan 1 2023

ASJC Scopus subject areas

  • General Mathematics

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