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      Refinements to the prime number theorem for arithmetic progressions

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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 and a refinement of Bombieri's "repulsive" log-free zero density estimate. Improvements exist when the modulus is sufficiently powerful.

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          Author and article information

          Journal
          24 August 2021
          Article
          2108.10878
          9567c374-c166-47e2-a287-0368a465fcc5

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          History
          Custom metadata
          11N05 (Primary), 11N13, 11M06 (Secondary)
          19 pages
          math.NT

          Number theory
          Number theory

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