A minor glitch in the proof of Lemma 6.3 is fixed, giving a slightly worse error term (changes in red). This does not affect any theorems. References updated
We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$. Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.
Most prime gaps results have been proven using tools from analytic or algebraic number theory in the last few centuries. In this paper, we would like to present some probabilistic way of proving many essential results. A major component of the proof is a probabilistic approach to the sieve method. In addition, we discuss their connections with recent work by Zhang and Maynard on small and large gaps in prime numbers.
Erdős considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments. Furthermore, we prove a generalisation stating that the gaps between primes $p$ for which there is no $\mathbb{Q}_p$-point on a random variety are Poisson distributed.
We show that a positive proportion of all gaps between consecutive primes are small gaps. We provide several quantitative results, some unconditional and some conditional, in this flavour.
We discuss recent advances on weak forms of the Prime $k$-tuple Conjecture, and its role in proving new estimates for the existence of small gaps between primes and the existence of large gaps between primes.
We present tables of record (maximal) gaps between densest prime constellations, or k-tuplets. The tables contain all maximal gaps between prime k-tuplets up to 10^15, for each k<=7.
We prove that a positive proportion of the gaps between consecutive primes are short gaps of length less than any fixed fraction of the average spacing between primes.
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…
We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.
In the framework of Cramer's probabilistic model of primes, we explore the exact and asymptotic distributions of maximal prime gaps. We show that the Gumbel extreme value distribution exp(-exp(-x)) is the limit law for maximal gaps between Cramer's random primes. The result can be derived from a general theorem about intervals between discrete random events occurring with slowly varying probability monotonically decreasing to zero. A straightforward generalization extends the Gumbel limit law to maximal gaps between prime constellations in Cramer's model.
We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.
In the present work we prove a common generalization of Maynard-Tao's recent result about consecutive bounded gaps between primes and on the Erdős-Rankin bound about large gaps between consecutive primes. The work answers in a strong form a 60 years old problem of Erdös, which asked whether the ratio of two consecutive primegaps can be infinitely often arbitrarily small, and arbitrarily large, respectively.
For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$ where $p$ is the smaller of the two primes.
In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this direct approach is a consequence of the the Kummer's characterization of summable sequences of positive terms. Some interesting consequences are then presented. In particular, we show how the twin-prime conjecture is related to our main result.
We derive heuristically formula for the $k$--moments $M_k(x)$ of the gaps between consecutive primes$<x $ represented directly by $x$$π(x)$ --- the number of primes up to: $M_k(x)= Γ(k+1)x^k/π^{k-1}(x)+\mathcal{O}(x)$, We illustrate obtained results by computer data.
We survey some past conditional results on the distribution of large differences between consecutive primes and examine how the Hardy-Littlewood prime k-tuples conjecture can be applied to this question.
Let $r \ge 2$ be an integer. We adapt the Maynard-Tao sieve to produce the asymptotically best-known bounded gaps between products of $r$ distinct primes. Our result applies to positive-density subsets of the primes that satisfy certain equidistribution conditions. This improves on the work of Thorne and Sono.
We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive this conclusion if there are certain types of exceptional zeros of Dirichlet L-functions.
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…
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. By studying this recursion on the cycles of gaps across stages of Eratosthenes sieve, we are able to provide evidence on a number of open problems regarding gaps between prime numbers. The basic counts of short constellations in the cycles of gaps provide evidence toward the twin prime conjecture and toward resolving a series of q…
In this paper, we apply the method of Maynard and Tao to the set of products of two distinct primes (E2-numbers). We obtain several results on the distribution of E2-numbers and primes. Among others, the result of Goldston, Pintz, Yildirim and Graham on small gaps between m consecutive E2-numbers is improved.
Combining the Hardy-Littlewood k-tuple conjecture with a heuristic application of extreme-value statistics, we propose a family of estimator formulas for predicting maximal gaps between prime k-tuples. Computations show that the estimator a(log(x/a)-b) satisfactorily predicts the maximal gaps below x, where a is the expected average gap between the same type of k-tuples, a=O(log^k x). Heuristics suggest that maximal gaps between prime k-tuples near x are approximately a*log(x/a), and thus have the order O(log^{k+1}x). The distribution of maximal gaps around the trend curve a*log(x/a) is close …
We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction. This answers a well-known question of Erdos.
As a refinement of the celebrated recent work of Yitang Zhang we show that any admissible k-tuple of integers contains at least two primes and almost primes in each component infinitely often if k is at least 181000. This implies that there are infinitely many gaps between consecutive primes of size at most 2530338, with an improved admissible k-tuple of Andrew W. Sutherland to 2326476.
For each $m\geq 1$, there exist infinitely many primes $p_1<p_2<\ldots<p_{m+1}$ such that $p_{m+1}-p_1=O(m^4e^{8m})$ and $p_j+2$ has at most $\frac{16m}{\log 2}+\frac{5\log m}{\log 2}+37$ prime divisors for each $j$.
Using Duke's large sieve inequality for Hecke Gr{ö}ssencharaktere and the new sieve methods of Maynard and Tao, we prove a general result on gaps between primes in the context of multidimensional Hecke equidistribution. As an application, for any fixed $0<ε<\frac{1}{2}$, we prove the existence of infinitely many bounded gaps between primes of the form $p=a^2+b^2$ such that $|a|<ε\sqrt{p}$. Furthermore, for certain diagonal curves $\mathcal{C}:ax^α+by^β=c$, we obtain infinitely many bounded gaps between the primes $p$ such that $|p+1-\#\mathcal{C}(\mathbb{F}_p)|<ε\sqrt{p}$.
We use Maynard's methods to show that there are bounded gaps between primes in the sequence $\{\lfloor nα\rfloor\}$, where $α$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties described by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $\lfloor f(q)\rfloor$ with $q$ a positive integer.
We study the first occurrences of gaps between primes in the arithmetic progression (P): $r$, $r+q$, $r+2q$, $r+3q,\ldots,$ where $q$ and $r$ are coprime integers, $q>r\ge1$. The growth trend and distribution of the first-occurrence gap sizes are similar to those of maximal gaps between primes in (P). The histograms of first-occurrence gap sizes, after appropriate rescaling, are well approximated by the Gumbel extreme value distribution. Computations suggest that first-occurrence gaps are much more numerous than maximal gaps: there are $O(\log^2 x)$ first-occurrence gaps between primes in (P) …
We generalise Zhang's and Pintz recent results on bounded prime gaps to give a lower bound for the the number of prime pairs bounded by 6*10^7 in the short interval $[x,x+x (\log x)^{-A}]$. Our result follows only by analysing Zhang's proof of Theorem 1, but we also explain how a sharper variant of Zhang's Theorem 2 would imply the same result for shorter intervals.
Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $[x-x^{0.525},x]$ for large $x$. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to intervals of this length. More precisely, we prove that for any $δ\in [0.525,1]$ there exist positive integers $k,d$ such that for sufficiently large $x$, the interval $[x-x^δ,x]$ contains $\gg_{k} \frac{x^δ}{(\log x)^k}$ pairs of consecutive primes differing by at most $d$. This confirms a speculation of Maynard that results on small gaps between primes can be re…
Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a fixed positive constant, for which the interval $(p_n, p_{n+1})$ contains the $k$-th power of a prime number.
We derive heuristically approximate formulas for the negative $k$--moments $M_{-k}(x)$ of the gaps between consecutive primes$<x $ represented directly by $π(x)$ --- the number of primes up to $x$. In particular we propose an analytical formula for the sum of reciprocals of gaps between consecutive primes $<x : ~ M_{-1}(x)\sim \frac{π^2(x)}{x-2π(x)}\log\Big(\frac{x}{2π(x)}\Big) \sim x \log \log(x)/\log^2(x)$. We illustrate obtained results by the enormous computer data up to $x=4\times 10^{18}$.
Suppose that $1<c<9/8$. For any $m\geq 1$, there exist infinitely many $n$ such that $$ \{[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]\} $$ contains at least $m+1$ primes, if $k_0$ is sufficiently large (only depending on $m$).
Question 10208b (1992) of the American Mathematical Monthly asked: does there exist an increasing sequence $\{a_k\}$ of positive integers and a constant $B > 0$ having the property that $\{ a_k + n\}$ contains no more than $B$ primes for every integer $n$? A positive answer to this question became known as Golomb's conjecture. In this note we give a negative answer, making use of recent progress in prime number theory.
We prove an analogue of a result by Goldston, Pintz and Yildirim for small gaps between primes that split completely in an abelian number field. We prove both a conditional result assuming the Elliott-Halberstam conjecture, and an unconditional result. We also give another proof of the same result in the special case of a quadratic extension of class number 1, which relies on a generalization of the Bombieri-Vinogradov theorem for quadratic number fields.
A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes $p_1,p_2$ with $|p_1-p_2|\leq 600$ as a consequence of the Bombieri-Vinogradov Theorem. In this paper, we apply his general method to the setting of Chebotarev sets of primes. We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over $\mathbb{Q}$, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms.
We introduce a refinement of the GPY sieve method for studying prime $k$-tuples and small gaps between primes. This refinement avoids previous limitations of the method, and allows us to show that for each $k$, the prime $k$-tuples conjecture holds for a positive proportion of admissible $k$-tuples. In particular, $\liminf_{n}(p_{n+m}-p_n)<\infty$ for any integer $m$. We also show that $\liminf(p_{n+1}-p_n)\le 600$, and, if we assume the Elliott-Halberstam conjecture, that $\liminf_n(p_{n+1}-p_n)\le 12$ and $\liminf_n (p_{n+2}-p_n)\le 600$.
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,…
A positive integer is called an $E_j$-number if it is the product of $j$ distinct primes. We prove that there are infinitely many triples of $E_2$-numbers within a gap size of $32$ and infinitely many triples of $E_3$-numbers within a gap size of $15$. Assuming the Elliot-Halberstam conjecture for primes and $E_2$-numbers, we can improve these gaps to $12$ and $5$, respectively. We can obtain even smaller gaps for almost primes, almost prime powers, or integers having the same exponent pattern in the their prime factorizations. In particular, if $d(x)$ denotes the number of divisors of $x$, we…
Let $α, β\geq 0$ and $α+ β< 1$. In this short note, we show that $\liminf_{n \to \infty} p_n^β(p_{n+1}^α- p_n^α) = 0$, where $p_n$ is the $n$th prime. This notes an improvement over results of Sándor and gives additional evidence towards a conjecture of Andrica. This follows directly from recent results on prime pairs from Maynard, Tao, Zhang.
Assuming the Riemann hypothesis, this article discusses a new elementary argument that seems to prove that the maximal prime gap of a finite sequence of primes p_1, p_2, ..., p_n <= x, satisfies max {p_(n+1) - p_n : p_n <= x} <= c1((logx)^2)/loglogx, c1 > 0 constant. Equivalently, it shows that the very short intervals (x, x + y] contain prime numbers for all y > c2((logx)^2)/loglogx, c2 > 0 constant, and sufficiently large x > 0.
Let $p_n$ denote the $n$th prime and $g_n:=p_{n+1}-p_n$ the $n$th prime gap. We demonstrate the existence of infinitely many values of $n$ for which $g_n>g_{n+1}>\cdots>g_{n+m}$ with $m\gg \log\log\log n$ and similarly for the reversed inequalities. In doing so we settle a conjecture of Erdös for the case $m=2$.
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 $\mathbb{F}_q(t)$.
Based on symbolic dynamics methods, we show the primes gap pattern could be described by the chaos orbit of Logistic mapping X(k+1)=1-uX(k)^2, u=1.5437. If so, there will be arbitrarily many twin primes.
A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer $g$ is a primitive root, provided $g \neq -1$ and $g$ is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide …
