Under GRH alone, square-root-weighted prime counts in residue classes satisfy the Chebyshev-bias asymptotic -M(a,b) log log x + C on a set of x of natural density 1.
Conrad, Partial Euler products on the critical line , Canad
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Chebyshev's bias without linear independence
Under GRH alone, square-root-weighted prime counts in residue classes satisfy the Chebyshev-bias asymptotic -M(a,b) log log x + C on a set of x of natural density 1.