相关论文: Negative moments of the gaps between consecutive p…

We derive heuristically formula for the $k$--moments $M_k(x)$ of the gaps between consecutive primes$<x $ represented directly by $x$$\pi(x)$ --- the number of primes up to: $M_k(x)= \Gamma(k+1)x^k/\pi^{k-1}(x)+\mathcal{O}(x)$, We…

For a real number $k$, define $\pi_k(x) = \sum_{p\le x} p^k$. When $k>0$, we prove that $$ \pi_k(x) - \pi(x^{k+1}) = \Omega_{\pm}\left(\frac{x^{\frac12+k}}{\log x} \log\log\log x\right) $$ as $x\to\infty$, and we prove a similar result when…

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…

A sharp asymptotic formula for the sum of reciprocals of $\pi(n)$ is derived, where $\pi(x)$ is the number of primes not exceeding $x$. This result improves the previous results of De Koninck--Ivi\'c and L. Panaitopol.

Let $p_{k}$ denote the $k$-th prime and $d(p_{k}) = p_{k} - p_{k - 1}$, the difference between consecutive primes. We denote by $N_{\epsilon}(x)$ the number of primes $\leq x$ which satisfy the inequality $d(p_{k}) \leq (\log p_{k})^{2 +…

We prove that given $\lambda \in \mathbb{R}$ such that $0 < \lambda < 1$, then $\pi(x + x^\lambda) - \pi(x) \sim \displaystyle \frac{x^\lambda}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short…

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…

We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a…

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 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 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…

The set of short intervals between consecutive primes squared has the pleasant---but seemingly unexploited---property that each interval $s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$ is fully sieved by the $k$ first primes. Here we take advantage of…

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…

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…

For $k\ge1$, let $R_k(x)$ denote the reciprocal sum up to $x$ of numbers with $k$ prime factors, counted with multiplicity. In prior work, the authors obtained estimates for $R_k(x)$, extending Mertens' second theorem, as well as a…

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…

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.

An overview of the results of new exhaustive computations of gaps between primes in arithmetic progressions is presented. We also give new numerical results for exceptionally large least primes in arithmetic progressions.

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.

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…