Highly Uniform Prime Number Theorems

TL;DR

This paper establishes a highly uniform prime number theorem for a broad class of $L$-functions, including Rankin-Selberg functions, with explicit error terms and polynomial ranges depending on the analytic conductor.

Contribution

It provides the first uniform prime number theorem results for Rankin-Selberg $L$-functions with explicit error bounds and polynomial ranges based on the analytic conductor.

Findings

Proved a uniform prime number theorem for a broad class of $L$-functions.

Derived explicit error terms depending on zero-free regions.

Established polynomial ranges in $x$ related to the analytic conductor.

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 $L(s,\pi \times \pi')$ associated to cuspidal automorphic representations $\pi$ and $\pi'$ of $\mathrm{GL}_{m}$ and $\mathrm{GL}_{m'}$, respectively. Our main result implies the first uniform prime number theorems for such $L$-functions (with analytic conductor uniformity) in complete generality.