Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim

Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture

In the present work we prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent fantastic achievement of Yitang Zhang about the existence of bounded gaps between consecutive primes. Most of these results would have belonged to the category of science fiction a decade ago. However, the presented results are far from being immediate consequences of Zhang's famous theorem: they require various new ideas, other important properties of the applied sieve functi…

The existence of small prime gaps in subsets of the integers

We consider the problem of finding small prime gaps in various sets of integers $\mathcal{C}$. Following the work of Goldston-Pintz-Yildirim, we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Letting $q_n$ denote the $n$-th prime in $\mathcal{C}$, we will establish that for any small constant $ε>0$, the set $\left\{q_n| q_{n+1}-q_n \leq ε\log n \right\}$ constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao we will also demonstrate that $\mathcal{C}$ has bounded prime gaps. Specific examples,…

Three topics in additive prime number theory

This is an expository article to accompany my two lectures at the CDM conference. I have used this an excuse to make public two sets of notes I had lying around, and also to put together a short reader's guide to some recent joint work with T.Tao. Contents: 1. An exposition, without much detail, of the work of Goldston, Pintz and Yildirim on gaps between primes; 2. A detailed discussion of the work of Mauduit and Rivat establishing that 50 percent of the primes have odd digit sum when written in base 2; 3. A reader's guide to recent work of T.Tao and the author on linear equations in primes. T…

Small gaps between configurations of prime polynomials

We find arbitrarily large configurations of irreducible polynomials over finite fields that are separated by low degree polynomials. Our proof adapts an argument of Pintz from the integers, in which he combines the methods of Goldston-Pintz-Yıldırım and Green-Tao to find arbitrarily long arithmetic progressions of generalized twin primes.

通过混沌多维筛与随机矩阵理论得到的启发式有界素数间隙

Heuristic Bounded Prime Gaps via a Chaotic Multidimensional Sieve and Random Matrix Theory

本文提出融合混沌扰动与RMT的EMCHS概率筛框架，在GPY与Maynard筛法基础上启发式改进素数间隙界，并用数值证据与部分严格证明加以支撑。

关于素数间大间隙的近期结果

Recent results on large gaps between primes

本文综述了素数间大间隙的近期研究成果，包括Ford等人的改进及Erdős等人的方法，并介绍了结合Maier矩阵方法研究大间隙中素数$k$次幂出现的新结果。

On the singular series in the prime k-tuple conjecture

In the present work a new simple proof of the theorem of Gallagher about the average of the singular series in the Hardy-Littlewood prime k-tuple conjecture is proved (in an even stronger form) which is uniform with respect to k (if the length of the interval $H$ is sufficiently large as a function of $k$). This result of Gallagher played a key role in our original joint work with D. A. Goldston and C. Y. Yildirim, where we showed the existence of infinitely many small gaps between consecutive primes. In the present work some weaker variants of Gallagher`s result are also proved (in an even ea…

On small gaps between primes and almost prime powers

In a recent joint work with D.A. Goldston and C.Y. Yildirim we just missed by a hairbreadth a proof that bounded gaps between primes occur infinitely often. In the present work it is shown that adding to the primes a much thinner set, called almost prime powers, the union of the set of primes and almost prime powers contains already infinitely many bounded gaps. More precisely it is shown that if we add to the set of primes either almost prime squares having exactly two, nearly equal prime factors or if we add to the set of primes almost prime cubes having exactly three, nearly equal prime fac…

Small Gaps between Primes Exist

In the recent preprint [3], Goldston, Pintz, and Yıldırım established, among other things, $$ \liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}=0,\leqno(0) $$ with $p_n$ the $n$th prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in [3]. While [3] also includes quantitative versions of $(0)$, we are concerned here solely with proving the qualitative $(0)$, which still exhibits all the essentials of the method. We also show here that an improvement of the Bombieri--Vinogradov prime number theorem would give rise infinitely …

The primes contain arbitrarily long arithmetic progressions

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi's theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemeredi's theorem that any subset of a sufficiently pseudorandom set of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yildirim. Using this…

A note on the distribution of normalized prime gaps

Let us denote the nth difference between consecutive primes by d_n. The Prime Number Theorem clearly implies that d_n is logn on average. Paul Erdős conjectured about 60 years ago that the sequence d_n/logn is everywhere dense on the nonnegative part of the real line. He and independently G. Ricci proved in 1954-55 that the set J of limit points of the sequence {d_n/logn} has positive Lebesgue measure. The first and until now only concrete known element of J was proved to be the number zero in the work of Goldston, Yildirim and the present author. The author of the present note showed in 2013 …