Refinements to the prime number theorem for arithmetic progressions

We prove a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel–Walfisz theorem, Hoheisel’s asymptotic for intervals of length \(x^{7/12+\varepsilon }\) $x^{7/12+\epsilon }$ , a Brun–Titchmarsh bound, and Linnik’s bound on the least prime in an arithmetic progression as corollaries. Our proof uses the Vinogradov–Korobov zero-free region, a log-free zero density estimate, and the Deuring–Heilbronn zero repulsion phenomenon. Improvements exist when the modulus is sufficiently powerful.