Refinements to the prime number theorem for arithmetic progressions

Jesse Thorner; Asif Zaman

doi:10.1007/s00209-023-03414-3

Refinements to the prime number theorem for arithmetic progressions

Jesse Thorner
, Asif Zaman

Mathematics

Research output: Contribution to journal › Article › peer-review

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^7/12+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.

Original language	English (US)
Article number	54
Journal	Mathematische Zeitschrift
Volume	306
Issue number	3
Early online date	Feb 20 2024
DOIs	https://doi.org/10.1007/s00209-023-03414-3
State	Published - Mar 2024

Keywords

Hoheisel’s theorem
Linnik’s theorem
Log-free zero density estimate
Primes in arithmetic progressions
Zero repulsion

ASJC Scopus subject areas

General Mathematics

Online availability

10.1007/s00209-023-03414-3

Library availability

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Cite this

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title = "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{\textquoteright}s asymptotic for intervals of length x7/12+ε, a Brun–Titchmarsh bound, and Linnik{\textquoteright}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.",

keywords = "Hoheisel{\textquoteright}s theorem, Linnik{\textquoteright}s theorem, Log-free zero density estimate, Primes in arithmetic progressions, Zero repulsion",

author = "Jesse Thorner and Asif Zaman",

note = "We thank Kevin Ford, Andrew Granville, Kannan Soundararajan, and Joni Ter{\"a}v{\"a}inen for their encouragement and helpful conversations as well as the anonymous referee for helpful comments. We especially thank Roger Heath-Brown for his comments regarding the proof of Theorem 2.1.",

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T1 - Refinements to the prime number theorem for arithmetic progressions

AU - Thorner, Jesse

AU - Zaman, Asif

N1 - We thank Kevin Ford, Andrew Granville, Kannan Soundararajan, and Joni Teräväinen for their encouragement and helpful conversations as well as the anonymous referee for helpful comments. We especially thank Roger Heath-Brown for his comments regarding the proof of Theorem 2.1.

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

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DO - 10.1007/s00209-023-03414-3

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Refinements to the prime number theorem for arithmetic progressions

Abstract

Keywords

ASJC Scopus subject areas

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