Chebyshev's bias without linear independence

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.