Related papers: Distribution of Large Gaps Between Primes

We investigate the behaviour of a certain additive function depending on prime divisors of specific integers lying in large gaps between consecutive primes. The result is obtained by a combination of results and ideas related to large gaps…

I present a new property of prime numbers that leads to a generalization of Cramer's conjecture. The study of the gap between consecutive primes is treated as a special case of the gap between consecutive terms of sequences having a certain…

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.

We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes,…

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 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…

We studied two probabilistic models of the distribution of primes in the natural number [1].The paper considers the third probabilistic model of the distribution of primes in the natural number. The author proved that the results obtained…

This is a survey article on the Hardy-Littlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture.

We prove that there are infinitely often pairs of primes much closer than the average spacing between primes - almost within the square root of the average spacing. We actually prove a more general result concerning the set of values taken…

We prove an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields.

Assuming a $q$-variant of the prime $k$-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions, respectively. This involves estimating the mean of singular series along…

We bring to bear an empirical model of the distribution of twin primes and produce two distinct results. The first is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the…

The Piatetski-Shapiro sequences are of the form ${\mathcal{N}}^{(c)} := (\lfloor n^c \rfloor)_{n=1}^\infty$ with $c > 1, c \not\in \mathbb{N}$. In this paper, we study the distribution of pairs $(p, p^{\#})$ of consecutive primes such that…

This paper demonstrates that from the Cramer's, Hardy-Littlewood's and Bateman-Horn's conjectures (suggest that the probability of a large positive integer being $x$ a prime - $\frac {1} {\ln(x)}$) it follows that the events consisting in a…

We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $\pi(x)$. As the application of this formula we formulate 7…

Using Duke's large sieve inequality for Hecke Gr{\"o}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,…

We establish, utilizing the Hardy-Littlewood Circle Method, an asymptotic formula for the number of pairs of primes whose differences lie in the image of a fixed polynomial. We also include a generalization of this result where differences…

In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples. In 2011 Wolf computed the "Skewes number" for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood…

For any real $x$ and any integer $k\ge1$, we say that a set $\mathcal{D}_{k}$ of $k$ distinct integers is a $k$-tuple jumping champion if it is the most common differences that occurs among $k+1$ consecutive primes less than or equal to…

For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $\varphi(q)$ reduced…