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arXiv · 2108.10878

Refinements to the prime number theorem for arithmetic progressions

Jesse Thorner·Asif Zaman
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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 $x^{1-\delta}$, 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.

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Jesse Thorner, Asif Zaman. 2021-08-24. Refinements to the prime number theorem for arithmetic progressions. https://doi.org/10.1007/s00209-023-03414-3

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