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Chebyshev's bias without linear independence

Mounir Hayani

TL;DR

This work addresses Chebyshev's bias in primes within residue classes using a weighted prime count $\pi_{1/2}(x;q,a)=\sum_{p\le x} p^{-1/2}$, without assuming linear independence of zeros. By employing GRH, explicit formulas, and a summation-by-parts approach, it proves that the natural and logarithmic densities for the weighted bias exist and equal 1, with a precise main term involving $\log\log x$ and a constant $C$, and a bias constant $M(q;a,b)$. In the special case $q=4$, this yields a natural density of 1 for $\pi_{1/2}(x;4,3) > \pi_{1/2}(x;4,1)$, demonstrating Chebyshev’s bias under GRH without LI or DRH. The results also establish DRH-type conclusions for almost all $x$ and provide a mean-value expression with an explicit leading term and tight error control, strengthening connections between classical prime number theorems in AP and modern bias phenomena.

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 related to L-function zeros that is stronger than GRH.

Chebyshev's bias without linear independence

TL;DR

This work addresses Chebyshev's bias in primes within residue classes using a weighted prime count , without assuming linear independence of zeros. By employing GRH, explicit formulas, and a summation-by-parts approach, it proves that the natural and logarithmic densities for the weighted bias exist and equal 1, with a precise main term involving and a constant , and a bias constant . In the special case , this yields a natural density of 1 for , demonstrating Chebyshev’s bias under GRH without LI or DRH. The results also establish DRH-type conclusions for almost all and provide a mean-value expression with an explicit leading term and tight error control, strengthening connections between classical prime number theorems in AP and modern bias phenomena.

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 related to L-function zeros that is stronger than GRH.
Paper Structure (3 sections, 6 theorems, 48 equations)

This paper contains 3 sections, 6 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of the main theorems

Key Result

Theorem 1.1

Let $q\ge 3$ and assume GRH for all non-principal Dirichlet characters modulo $q$. Let $\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\, a,$ and $b$, such that the natural density of the set exists and equals $1$, and the logarithmic density of the set exists and equals $1$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['main']}
  • proof : Proof of Theorem \ref{['GRHalmimpliesDRH']}
  • ...and 1 more