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Primes in arithmetic progressions to large moduli II: Well-factorable estimates

James Maynard

Abstract

We establish new mean value theorems for primes of size $x$ in arithmetic progressions to moduli as large as $x^{3/5-ε}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most $x^{4/7-ε}$. This has consequences for the level of distribution for sieve weights coming from the linear sieve.

Primes in arithmetic progressions to large moduli II: Well-factorable estimates

Abstract

We establish new mean value theorems for primes of size in arithmetic progressions to moduli as large as when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most . This has consequences for the level of distribution for sieve weights coming from the linear sieve.

Paper Structure

This paper contains 9 sections, 23 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Proof outline
  3. Acknowledgements
  4. Notation
  5. Proof of Theorem \ref{['thrm:Factorable']}
  6. Preparatory lemmas
  7. Double divisor function estimates
  8. Well-factorable estimates
  9. Proof of Theorem \ref{['thrm:Linear']}

Key Result

Theorem A

Let $a\in\mathbb{Z}$ and $A,\epsilon>0$. Let $\lambda_q$ be a sequence which is well-factorable of level $Q\le x^{4/7-\epsilon}$. Then we have

Theorems & Definitions (28)

  • Definition 1: Well factorable
  • Theorem A: Bombieri, Friedlander, Iwaniec
  • Definition 2: Triply well factorable
  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Definition 3: Siegel-Walfisz condition
  • Proposition 5.1: Well-factorable Type II estimate
  • Proposition 5.2: Divisor function in arithmetic progressions
  • Lemma 5.3: Heath-Brown identity
  • ...and 18 more