Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Refinements to the prime number theorem for arithmetic progressions

Jesse Thorner, Asif Zaman

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 $x^{1-δ}$, 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.

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 , 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.

Paper Structure

This paper contains 8 sections, 8 theorems, 57 equations.

Table of Contents

  1. Introduction and statement of results
  2. Main result
  3. Connections with classical results
  4. Proof of \ref{['thm:main']}
  5. Further refinements
  6. Powerful moduli
  7. Comparison with work of Gallagher
  8. A flexible prime number theorem: Proof of \ref{['thm:FlexiblePNT']}

Key Result

Theorem 1.1

Let $q\geq 2$ and $a$ be coprime integers and $4 \leq h \leq x$. Define $\lambda$ and $\theta$ by For all $0<\varepsilon<1-\theta$, there is a constant $\Cl[abcon]{main}=\Cr{main}(\varepsilon)>0$ such that if $\lambda h/\varphi(q) \geq x^{\theta+\varepsilon}$, then The implied constant and $\Cr{main}$ are effectively computable, and $\log^+(u) = \max\{0,\log u\}$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 2.1
  • Remark 2.2
  • proof
  • Theorem 2.3
  • Remark 2.4
  • ...and 8 more