Related papers: Short intervals containing a prescribed number of …

This note presents a result on the maximal prime gap of the form p_(n+1) - p_n <= C(log p_n)^(1+e), where C > 0 is a constant, for any arbitrarily small real number e > 0, and all sufficiently large integer n > n_0. Equivalently, the result…

We show that there exist real numbers $\alpha_1,\alpha_2$ linearly independent over $\mathbb{Z}$ together with 1 such that for every non-zero integer vector $(m_1,m_2)$ with $m_1\ge 0$ and $m_2\ge 0$ one has $||m_1\alpha_1+m_2\alpha_2|| \ge…

It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that there is a prime between consecutive cubes…

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 Erd\H{o}s conjectured about 60 years ago that the sequence d_n/logn is everywhere dense on the…

Goldston, Pintz and Y\i ld\i r\i m have shown that if the primes have `level of distribution' $\theta$ for some $\theta>1/2$ then there exists a constant $C(\theta)$, such that there are infinitely many integers $n$ for which the interval…

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then there are at least $n$ primes in the interval $(x/2,x]$. For example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists and…

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…

Let $x$ be a positive real number, and $\mathcal{P} \subset [2,\lambda(x)]$ be a set of primes, where $\lambda(x) \in \Omega(x^\varepsilon)$ is a monotone increasing function with $\varepsilon \in (0,1)$. We examine $Q_{\mathcal{P}}(x)$,…

Let $A\subset [1,x]$ be a non-empty set of primes with $|A|= \alpha x(\log x)^{-1}$. We prove that there exist absolute constants $c_1,c_2>0$ such that, as $x$ gets sufficiently large, we have $|A+A|\geq c_1(\log x)(\log \log…

Under Cram\'er's conjecture concerning the prime numbers, we prove that for any $x>1$, there exists a real $A=A(x)>1$ for which the formula $[A^{n^x}]$ (where $[]$ denotes the integer part) gives a prime number for any positive integer $n$.…

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…

Given a zero-free region and an averaged zero-density estimate over all Dirichlet $L$-functions modulo $q\in\mathbb{N}$, we refine the error terms of the prime number theorem in all and almost all short arithmetic progressions. For example,…

Let $S = \{q_1, \ldots , q_s\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m = q_1^{r_1} \ldots q_s^{r_s} M$, where $r_1, \ldots , r_s$ are non-negative integers and $M$ is an integer relatively…

Let $n\in\mathbb{Z}^+$. In [8] we ask the question whether any sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number, and we show that this is actually the case for every…

We show that at least 1/3 of positive real numbers are in the set of limit points of normalized prime gaps. More precisely, if $p_n$ denotes the $n$th prime and $\mathbb{L}$ is the set of limit points of the sequence $\{(p_{n+1}-p_n)/\log…

Let $p$ and $q$ be two distinct fixed prime numbers and $(n_i)_{i\geq 0}$ the sequence of consecutive integers of the form $p^a\cdot q^b$ with $a,b\ge 0$. Tijdeman gave a lower bound (1973) and an upper bound (1974) for the gap size…

Fix \epsilon > 0, and let p_1 = 2, p_2 = 3,... be the sequence of all primes. We prove that if (q,a) = 1 then there are infinitely many pairs p_r, p_{r+1} such that p_r \equiv p_{r+1} \equiv a \mod q and p_{r+1} - p_r < \epsilon\log p_r.…

Let $f(n)=\min_{p} |n-p|$, where $p$ is a prime. We show that there is a positive constant $\delta$ such that for any large integer $N$ there exist two positive integers $n_1$ and $n_2$ such that $N=n_1 + n_2$ and $f(n_i)\gg \ln N (\ln\ln…

A positive integer $n$ is called an abundant number if $\sigma (n)\ge 2n$, where $\sigma (n)$ is the sum of all positive divisors of $n$. Let $E(x)$ be the largest number of consecutive abundant numbers not exceeding $x$. In 1935, P. Erd\H…

Let $p\geq 2$ be a large prime, and let $N\gg ( \log p)^{1+\varepsilon}$. This note proves the existence of primitive roots in the short interval $[M,M+N]$, where $M \geq 2$ is a fixed number, and $ \varepsilon>0$ is a small number. In…