Highly Uniform Prime Number Theorems

Abstract

We prove a highly uniform version of the prime number theorem for a certain class of $L$-functions. The range of $x$ depends polynomially on the analytic conductor, and the error term is expressed in terms of an optimization problem depending explicitly on the available zero-free region. The class contains the Rankin-Selberg $L$-function $L(s,\pi \times {\pi }^{\prime })$ associated to cuspidal automorphic representations $\pi$ and ${\pi }^{\prime }$ of ${\mathrm{GL}}_m$ and ${\mathrm{GL}}_{m^{\prime }}$, respectively. Our main result implies the first uniform prime number theorems for such $L$-functions (with analytic conductor uniformity) in complete generality.

16 pages. Incorporates referee comments. Theorem 2.6 is improved

References in corpus (1)

• Towards a ${\mathrm{GL}}_n$ variant of the Hoheisel phenomenon