Chebyshev's bias without linear independence

TL;DR

This paper proves that primes are more abundant in certain residue classes under GRH without requiring assumptions beyond the hypothesis, using a novel density approach with prime weights.

Contribution

It establishes Chebyshev's bias in prime distribution without linear independence assumptions and without restricting to logarithmic densities, under GRH.

Findings

Prime counts in residue classes have density 1 when weighted by inverse square root.

Chebyshev's bias is confirmed without stronger zero hypotheses.

No need for linear independence or logarithmic density restrictions.

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.