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Primes in arithmetic progressions to large moduli III: Uniform residue classes

James Maynard

Abstract

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1/2+δ}$ which have conveniently sized divisors. The main feature of these estimates is that they are completely uniform with respect to the residue classes considered, unlike previous works on primes in arithmetic progressions to large moduli.

Primes in arithmetic progressions to large moduli III: Uniform residue classes

Abstract

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than , extending the Bombieri-Vinogradov theorem to moduli of size which have conveniently sized divisors. The main feature of these estimates is that they are completely uniform with respect to the residue classes considered, unlike previous works on primes in arithmetic progressions to large moduli.

Paper Structure

This paper contains 14 sections, 51 theorems, 398 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgements
  4. Notation
  5. Main Propositions
  6. Preparatory lemmas
  7. Proof of Theorem \ref{['thrm:WeakEquidistribution']}
  8. Proof of Theorem \ref{['thrm:AlmostUniform']}
  9. Proof of Theorem \ref{['thrm:Minorant']}
  10. Further preparatory lemmas
  11. Main Type II estimate
  12. Second Type II estimate
  13. Zhang-style Type II estimate
  14. Triple divisor function

Key Result

Theorem 1.1

Let $C>0$ be a sufficiently large absolute constant and $\delta>0$. Let $Q_1\le x^{1/10-3\delta}/(\log{x})^C$ and $Q_2\le x^{4/10+4\delta}(\log{x})^C$. Then we have

Theorems & Definitions (104)

  • Theorem 1.1: Uniform equidistribution of primes with weak error term
  • Theorem 1.2: Almost uniform equidistribution for primes
  • Theorem 1.3: Uniform equidistribution of a minorant
  • Corollary 1.4: Primes in all progressions for almost-all moduli
  • Remark
  • Remark
  • Definition 1: Siegel-Walfisz condition
  • Proposition 5.1: Type II estimate
  • Proposition 5.2: Type II estimate near $x^{2/5}$
  • Proposition 5.3: Type II estimate near $x^{1/2}$
  • ...and 94 more