Related papers: Primes in the intervals between primes squared

We show that for all large enough $x$ the interval $[x,x+x^{1/2}\log^{1.39}x]$ contains numbers with a prime factor $p > x^{18/19}.$ Our work builds on the previous works of Heath-Brown and Jia (1998) and Jia and Liu (2000) concerning the…

The author gives nontrivial upper and lower bounds for the number of primes in the interval $[x - x^{\theta}, x]$ for some $0.52 \leqslant \theta \leqslant 0.525$, showing that the interval $[x - x^{0.52}, x]$ contains prime numbers for all…

The results of the study provide guidelines for the development and applications of algorithms. When the number of steps for calculating an assumption tends to infinity, probability theory can be applied to predict whether the assumption…

For $n \geq 3,$ let $ p_n $ denote the $n^{\rm th}$ prime number. Let $[ \; ]$ denote the floor or greatest integer function. For a positive integer $m,$ let $\pi_2(m)$ denote the number of twin primes not exceeding $m.$ The twin prime…

We introduce the sequence $(a_n) \subset (0,1]$ and prove that the asymptotic behaviour of $\sum_{k=1}^n a_k$ is the same than $\pi(n)$, the prime-counting function. We also obtain that $\pi(n) \sim n a_n$ and we estimate…

In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vall\'ee Poussin's form: $$ \pi(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ Instead of performing asymptotic expansion on Chebyshev…

We prove that the average size of the squares of differences between consecutive primes less than $x$ is $O(x^{0.23+\varepsilon})$ for any fixed $\varepsilon>0$. This improves on a result of Peck, who gave bound $O(x^{0.25+\varepsilon})$ in…

While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the…

An open conjecture of Z.-W. Sun states that for any integer $n>1$ there is a positive integer $k\le n$ such that $\pi(kn)$ is prime, where $\pi(x)$ denotes the number of primes not exceeding $x$. In this paper, we show that for any positive…

We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations.…

We prove a generalization of the author's work to show that any subset of the primes which is `well-distributed' in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the…

We are interested in classifying those sets of primes $\mathcal{P}$ such that when we sieve out the integers up to $x$ by the primes in $\mathcal{P}^c$ we are left with roughly the expected number of unsieved integers. In particular, we…

Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…

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…

Assuming the Riemann Hypothesis, we prove that for all $x\geq 2$, there exists at least one even integer within the interval $(x, x+123\log^2x]$, that can be expressed as the sum of two primes. This result is an improvement over the recent…

In this article, we prove an "equivalence" between two higher even moments of primes in short intervals under Riemann Hypothesis. We also provide numerical evidence in support of these asymptotic formulas.

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, the authors of a recent paper showed that the expected number of…

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll (\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$…

In this paper, the estimation formula of the number of primes in a given interval is obtained by using the prime distribution property. For any prime pairs $p>5$ and $ q>5 $, construct a disjoint infinite set sequence $A_1, A_2, \ldots,…

For $x\ge0$ let $\pi(x)$ be the number of primes not exceeding $x$. The asymptotic behaviors of the prime-counting function $\pi(x)$ and the $n$-th prime $p_n$ have been studied intensively in analytic number theory. Surprisingly, we find…