TY - JOUR
T1 - A Bombieri-Vinogradov Theorem for Higher-Rank Groups
AU - Jiang, Yujiao
AU - Lü, Guangshi
AU - Thorner, Jesse
AU - Wang, Zihao
N1 - Publisher Copyright:
© The Author(s) 2021. Published by Oxford University Press. All rights reserved.
PY - 2023/1/1
Y1 - 2023/1/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85151866946&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85151866946&partnerID=8YFLogxK
U2 - 10.1093/imrn/rnab261
DO - 10.1093/imrn/rnab261
M3 - Article
AN - SCOPUS:85151866946
SN - 1073-7928
VL - 2023
SP - 482
EP - 535
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 1
ER -