Weighted Prime Number Theorem on Arithmetic Progressions with Refinements

We present an extension of the Dirichlet-type prime number theorem to weighted counting functions, the importance of which has recently been recognized for formulating Chebyshev’s bias. Moreover, we prove that their difference πw(x;q,a)−πw(x;q,b) (0≤w<1/2) changes its sign infinitely many times as x grows for any coprime a,b(a≠b) with q, 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 w=1/2.