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On the frequency of small gaps between the primes

Akos Magyar, Janos Pintz

TL;DR

The paper proves an unconditional lower bound on the frequency of small prime gaps by adapting a Selberg-type weighted sieve to detect primes in admissible $k$-tuples and exploiting strong equidistribution results from Zhang–Polymath 8A. By carefully bounding weighted sums $S_0(\mathcal{H})$ and $S_1(\mathcal{H},h^*)$ and controlling almost-prime contributions, the authors translate sieve estimates into quantitative lower bounds for the number of prime gaps $\le \eta \log N$ up to $N$, obtaining a positive proportion for a wide range of gap sizes (up to about $c\log N$) with explicit exponents. The method yields an unconditional bound of the form $\#\{p\le N: p'-p\le \eta\log p\} \gg_{\eta} \pi(N)$ for certain small $\eta$, and in particular derives a lower bound of the form $c_0 \eta^{1879} \pi(N)$ for gaps of size at most $\eta \log N$, while acknowledging that modern refinements (Maynard–Polymath) could substantially reduce the exponent. This work links GPY-type sieve machinery with the Polymath 8A Bombieri–Vinogradov-type input to push unconditional results on small prime gaps forward.

Abstract

In a recent work Friedlander studied the problem of how large consecutive prime gaps should be in order that the sum of the reciprocals should be divergent. Supposing a very deep Hypothesis, a generalization of the Hardy--Littlewood prime $k$-tuple conjecture, he gave an almost precise answer for it. In the present work we give an unconditional answer for a much weaker form of the same problem.

On the frequency of small gaps between the primes

TL;DR

The paper proves an unconditional lower bound on the frequency of small prime gaps by adapting a Selberg-type weighted sieve to detect primes in admissible -tuples and exploiting strong equidistribution results from Zhang–Polymath 8A. By carefully bounding weighted sums and and controlling almost-prime contributions, the authors translate sieve estimates into quantitative lower bounds for the number of prime gaps up to , obtaining a positive proportion for a wide range of gap sizes (up to about ) with explicit exponents. The method yields an unconditional bound of the form for certain small , and in particular derives a lower bound of the form for gaps of size at most , while acknowledging that modern refinements (Maynard–Polymath) could substantially reduce the exponent. This work links GPY-type sieve machinery with the Polymath 8A Bombieri–Vinogradov-type input to push unconditional results on small prime gaps forward.

Abstract

In a recent work Friedlander studied the problem of how large consecutive prime gaps should be in order that the sum of the reciprocals should be divergent. Supposing a very deep Hypothesis, a generalization of the Hardy--Littlewood prime -tuple conjecture, he gave an almost precise answer for it. In the present work we give an unconditional answer for a much weaker form of the same problem.
Paper Structure (3 sections, 3 theorems, 60 equations)

This paper contains 3 sections, 3 theorems, 60 equations.

Table of Contents

  1. Main ideas of the proof. Notation
  2. Weighted sums of prime $k$-tuples
  3. Proof of Theorem 1

Key Result

Theorem 1

If $N$ is sufficiently large, that is $N > C_1$ (explicitly calculable), $\eta = \eta(N) > C_0/\log N$ ($C_0$ explicitly computable positive constant), then with an explicitly calculable $c_0 > 0$, where $n \sim N$ means $N \leq n < 2N$.

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Remark 1
  • Remark 2