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Primes in arithmetic progressions to large moduli I: Fixed residue classes

James Maynard

Abstract

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1/2+δ}$ with a 'convenient sized' factor. As a consequence, the expected asymptotic holds for all but $O(δQ)$ moduli $q\sim Q=x^{1/2+δ}$ and we get results for moduli as large as $x^{11/21}$. Our proof extends previous techniques of Bombieri, Fouvry, Friedlander and Iwaniec by incorporating new ideas inspired by amplification methods. We combine these with techniques of Zhang and Polymath tailored to our application. In particular, we ultimately rely on exponential sum bounds coming from the spectral theory of automorphic forms (the Kuznetsov trace formula) or from algebraic geometry (Weil and Deligne style estimates).

Primes in arithmetic progressions to large moduli I: Fixed residue classes

Abstract

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than . Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size with a 'convenient sized' factor. As a consequence, the expected asymptotic holds for all but moduli and we get results for moduli as large as . Our proof extends previous techniques of Bombieri, Fouvry, Friedlander and Iwaniec by incorporating new ideas inspired by amplification methods. We combine these with techniques of Zhang and Polymath tailored to our application. In particular, we ultimately rely on exponential sum bounds coming from the spectral theory of automorphic forms (the Kuznetsov trace formula) or from algebraic geometry (Weil and Deligne style estimates).

Paper Structure

This paper contains 27 sections, 72 theorems, 550 equations.

Table of Contents

  1. Introduction
  2. Comparison with previous results
  3. Acknowledgements
  4. Outline
  5. High-level overview
  6. Critical case in the Bombieri--Friedlander--Iwaniec argument
  7. The original BFI argument
  8. A modified argument
  9. Combining with ideas of Zhang
  10. Quantitative bounds
  11. Corollaries \ref{['crllry:BVDivisor']}, \ref{['crllry:MostModuli']} and \ref{['crllry:1121']} from Theorem \ref{['thrm:MainTheorem']}
  12. Paper structure
  13. Notation
  14. Sieve decomposition
  15. Type II estimates
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $a\in\mathbb{Z}$, let $\epsilon>0$ and let $Q_1,Q_2$ satisfy Then for every $A>0$ we have

Theorems & Definitions (74)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem A: Bombieri, Friedlander, Iwaniec
  • Theorem B: Bombieri, Friedlander, Iwaniec
  • Theorem C: Zhang, Polymath
  • Definition 1: Siegel-Walfisz condition
  • Proposition 7.1: Type II estimate
  • Proposition 7.2: Sieve asymptotics
  • ...and 64 more