Highly Uniform Prime Number Theorems

TL;DR

This work proves a highly uniform prime number theorem for a broad class of $L$-functions $\mathfrak{S}(m)$, including standard and Rankin–Selberg $L$-functions, by incorporating properties (A)–(F) and additional zero-free hypotheses. The authors develop a smoothing framework and leverage log-free zero-density estimates together with zero-repulsion phenomena to derive an explicit asymptotic for $\sum_{n\le x} a_{\pi}(n)\Lambda(n)$ with leading term $r_{\pi}x$ and a correction term $-x^{\beta_0}/\beta_0$, plus a precisely quantified error depending on $\eta_{\pi}(x)$ and the analytic conductor $C(\pi)$. They apply the result to standard and Rankin–Selberg $L$-functions, obtaining the first uniform PNTs with conductor-uniformity in complete generality for the non-self-dual Rankin–Selberg case, and providing comparisons with prior results such as those of Iwaniec–Kowalski and the Soundararajan–Thorner framework. The methods combine Hadamard factorization, smoothing, Mellin transforms, and log-free zero-density bounds, enabling explicit dependence on $C(\pi)$ and allowing optimization over parameters to maximize the range for which the prime number theorem holds with nontrivial error.

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 $L(s,π\times π')$ associated to cuspidal automorphic representations $π$ and $π'$ 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.

Theorems & Definitions (43)

• Remark
• Remark
• Remark
• Theorem 1.1
• Theorem 1.2: IK
• Lemma 2.1
• proof
• Proposition 2.2
• Proposition 2.3
• proof