Chebyshev's bias without linear independence

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.