Abstract:We confirm Chebyshev's observation that primes are strikingly more abundant in non-square residue classes modulo a fixed integer under the Generalized Riemann Hypothesis (GRH) by proving a (natural) density $1$ statement for prime counting functions in residue classes where each prime is weighted by its inverse square root. In contrast to the majority of the existing literature on the subject, we do not need to restrict to logarithmic densities to measure Chebyshev's bias, and we do not rely on any hypothesis on the zeros of $L$-functions that is stronger than GRH. Note: The same type of results presented here were independently proved by Arshay Sheth (2025) in the general context of automorphic forms, which implies our main asymptotic. While the spirit of the proofs is similar, Sheth develops an explicit formula for the partial Euler product and uses a result due to Gallagher (1980) to prove that some estimates hold outside a set of finite logarithmic measure. We share this independent work because it provides a completely self-contained and elementary proof relying only on the usual explicit formula, and it yields explicit error terms rather than an implicit $o(1)$ asymptotic.
From: Mounir Hayani [view email] [via CCSD proxy]
[v1]
Mon, 29 Dec 2025 08:44:00 UTC (9 KB)
[v2]
Mon, 5 Jan 2026 15:55:55 UTC (11 KB)
[v3]
Thu, 25 Jun 2026 11:35:33 UTC (12 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。