A Bombieri-Vinogradov theorem for higher rank groups

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 $\pi$ of $\mathrm{GL}_n$ over a number field $F$ which 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 $\pi$ 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 $|L(1,\pi\otimes\chi)|$ in the $\chi$-aspect, where $\chi$ 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 $\mathrm{GL}_n$ analogue of the Titchmarsh divisor problem and a nontrivial bound for a certain $\mathrm{GL}_n\times\mathrm{GL}_2$ shifted convolution sum.