Refinements to the prime number theorem for arithmetic progressions

Abstract. 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, 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 and a refinement of Bombieri’s “repulsive” log-free zero density estimate. Improvements exist when the modulus is sufficiently powerful.

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