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Bounded gaps between primes in number fields and function fields

Abel Castillo, Chris Hall, Robert J. Lemke Oliver, Paul Pollack, Lola Thompson

Abstract

The Hardy--Littlewood prime $k$-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field $\mathbb{F}_q(t)$.

Bounded gaps between primes in number fields and function fields

Abstract

The Hardy--Littlewood prime -tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field .

Paper Structure

This paper contains 8 sections, 14 theorems, 57 equations.

Table of Contents

  1. Introduction and statement of results
  2. The general Maynard--Tao method
  3. The dictionary
  4. Sieve manipulations: Multiplicative functionology
  5. Final assembly of theorems
  6. Applications
  7. Bounded gaps in totally real number fields
  8. Gap densities in $\mathbb{F}_q(t)$

Key Result

Theorem 1

Let $m\geq 2$. There exists a constant $k_0:=k_0(m)$ such that, for any admissible $k$-tuple $\mathcal{H}=(h_1,\dots,h_k)$ with $k\geq k_0$, there are infinitely many $n$ such that at least $m$ of $n+h_1,\dots,n+h_k$ are prime.

Theorems & Definitions (19)

  • Conjecture
  • Theorem : Maynard--Tao
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark
  • Theorem 1.4
  • Remark
  • Proposition 2.1
  • Lemma 2.2
  • ...and 9 more