TY - JOUR

T1 - Refinements to the prime number theorem for arithmetic progressions

AU - Thorner, Jesse

AU - Zaman, Asif

N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.

PY - 2024/3

Y1 - 2024/3

N2 - 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 x7/12+ε, 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.

AB - 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 x7/12+ε, 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.

KW - Hoheisel’s theorem

KW - Linnik’s theorem

KW - Log-free zero density estimate

KW - Primes in arithmetic progressions

KW - Zero repulsion

UR - http://www.scopus.com/inward/record.url?scp=85185456164&partnerID=8YFLogxK

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U2 - 10.1007/s00209-023-03414-3

DO - 10.1007/s00209-023-03414-3

M3 - Article

AN - SCOPUS:85185456164

SN - 0025-5874

VL - 306

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3

M1 - 54

ER -