Weighted Prime Number Theorem on Arithmetic Progressions with Refinements

We extend Dirichlet’s prime number theorem on arithmetic progressions to the weighted counting function πs(x; q, a) for (a, q) = 1 with the weight p−w for prime p with w ≥ 0. We also prove that when 0 ≤ w < 1/2, the difference πw(x; q, a) − πw(x; q, b) changes its sign infinitely many times as x grows for any coprime a, b (a ̸= b) with p under the assumption that Dirichlet L-functions have no real nontrivial zeros.. This result gives a justification of the theory of Aoki-Koyama that Chebyshev’s bias is formulated by the asymptotic behavior of πw(x; q, a) − πw(x; q, b) at s = 1/2.