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A modification of the linear sieve, and the count of twin primes

Jared Duker Lichtman

Abstract

We introduce a modification of the linear sieve whose weights satisfy strong factorization properties, and consequently equidistribute primes up to size $x$ in arithmetic progressions to moduli up to $x^{10/17}$. This surpasses the level of distribution $x^{4/7}$ with the linear sieve weights from well-known work of Bombieri, Friedlander, and Iwaniec, and which was recently extended to $x^{7/12}$ by Maynard. As an application, we obtain a new upper bound on the count of twin primes. Our method simplifies the 2004 argument of Wu, and gives the largest percentage improvement since the 1986 bound of Bombieri, Friedlander, and Iwaniec.

A modification of the linear sieve, and the count of twin primes

Abstract

We introduce a modification of the linear sieve whose weights satisfy strong factorization properties, and consequently equidistribute primes up to size in arithmetic progressions to moduli up to . This surpasses the level of distribution with the linear sieve weights from well-known work of Bombieri, Friedlander, and Iwaniec, and which was recently extended to by Maynard. As an application, we obtain a new upper bound on the count of twin primes. Our method simplifies the 2004 argument of Wu, and gives the largest percentage improvement since the 1986 bound of Bombieri, Friedlander, and Iwaniec.

Paper Structure

This paper contains 17 sections, 27 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Application to twin primes
  3. Outline of main ideas in Theorem \ref{['thm:etalinsieve']}
  4. Technical setup and results
  5. Factorization of weights and their level of distribution
  6. Sieve theory setup and bounds
  7. Programmably factorable support
  8. Refined factorization of $\mathcal{D}^{\textnormal{well}}$
  9. Modification of the linear sieve
  10. Sieve function computation
  11. Factorable remainder, after Iwaniec
  12. Variable level of distribution for the linear sieve weights
  13. Equidistribution for products of primes
  14. Upper bound for twin primes
  15. Lemmas
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $D = x^{\frac{10}{17}-{\varepsilon}}$. There exists a sequence $\widetilde{\lambda}^*(d)\in\{-1,0,1\}$ satisfying: (1) Equidistribution for primes: for any fixed $a\in{\mathbb Z}$, $A,{\varepsilon}>0$, we have (2) Sieve upper bound: for $s{\,\geqslant}1$, $z=D^{1/s}$, we have where $F^*(s) \, {\,\leqslant} \, 1.000081\,F(s)$ when $1{\,\leqslant} s{\,\leqslant} 3$, for the linear sieve funct

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: well-factorable
  • Theorem 2.2: Bombieri--Friedlander--Iwaniec BFI1
  • Definition 2.3: triply well-factorable
  • Definition 2.4: programmably factorable
  • Theorem 2.5: Maynard JM2
  • Remark 2.6
  • Corollary 2.7: Maynard JM2
  • Remark 2.8
  • ...and 42 more