Related papers: Numbers with at most $2$ prime factors in short ar…

We prove that there are infinitely many integers $n$ such that the total number of prime factors of $(n+h_{1})(n+h_{2})...(n+h_{\kappa})$ is at most $(1/2)\kappa\log\kappa+O(\kappa)$, provided $\kappa$ is sufficiently large.

Let $f(n)$ be the number of distinct exponents in the prime factorization of the natural number $n$. We prove some results about the distribution of $f(n)$. In particular, for any positive integer $k$, we obtain that $$ \#\{n \leq x : f(n)…

R. Hofer and A. Winterhof proved that the 2-adic complexity of the two-prime (binary) generator of period $pq$ with two odd primes $p\neq q$ is close to its period and it can attain the maximum in many cases. When the two-prime generator is…

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…

We prove an upper bound on the density of zeros very close to the critical line of the family of Dirichlet $L$-functions of modulus $q$ at height $T$. To do this, we derive an asymptotic for the twisted second moment of Dirichlet…

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1/2+\delta}$ with a…

Let $E/\mathbb Q$ be an elliptic curve, and denote by $N(p)$ the number of $\mathbb{F}_p$-points of the reduction modulo $p$ of $E$. A conjecture of Koblitz, refined by Zywina, states that the number of primes $p \leq X$ at which $N(p)$ is…

Let $p_n$ denote the $n^{th}$ prime. Goldston, Pintz, and Yildirim recently proved that $ \liminf_{n\to \infty} \frac{(p_{n+1}-p_n)}{\log p_n} =0.$ We give an alternative proof of this result. We also prove some corresponding results for…

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi's theorem, which asserts that any subset of the integers of positive density contains progressions of…

For any large prime $q$, $x \leq 1$ and any real $k\geq 2$, we prove a lower bound for the following $2k$-th moment \begin{equation*} \sum_{\substack{\chi \in X_q^*}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k}, \end{equation*} where…

Let $\alpha$ be a real number such that $1< \alpha <2$ and let $x_0=x_0(\alpha)$ be a {\rm(}unique{\rm)} positive solution of the equation $$ x^{\alpha-1} -\frac{\pi}{e^2\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer…

Let $S(n)$ denote the least primary factor in the primary decomposition of the multiplicative group $M_n = (\Bbb Z/n\Bbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/(\log x)^{1/2}$, for the counting function of…

This document seeks to prove there are infinitely many primes whose difference is 2, referred to as twin prime pairs. This proof's methodology involves constructing a function that approximates the number of positive integers, less than a…

Let $\lambda$ denote the Liouville function. Assuming the Riemann Hypothesis, we prove that $$\int_X^{2X}\Big|\sum_{x\leq n \leq x+h}\lambda(n) \Big|^2 dx \ll Xh(\log X)^6,$$ as $X\rightarrow \infty$, provided $h=h(X)\leq…

We prove lower bounds for the number of primes $p \leq N + b$ such that $p-b$ is divisible by $2^{k(N)}$ and has at most $k$ odd prime factors ($k \geq 2$), assuming $2^{k(N)} \leq N^\theta$ for some $\theta > 0$ depending on $k$. The proof…

Let $p \in (0, \infty)$ be a constant and let $\{\xi_n\} \subset L^p(\Omega, {\mathcal F}, \P)$ be a sequence of random variables. For any integers $m, n \ge 0$, denote $S_{m, n} = \sum_{k=m}^{m + n} \xi_k$. It is proved that, if there…

Via (C,F)-construction we produce a 2-fold simple mixing transformation which has uncountably many non-trivial proper factors and all of them are prime.

Let $q>r\ge1$ be coprime integers. Let $R(n,q,r)$ be the $n$th record gap between primes in the arithmetic progression $r$, $r+q$, $r+2q,\ldots,$ and denote by $N_{q,r}(x)$ the number of such records observed below $x$. For $x\to\infty$, we…

We prove that for all $n\geq 1$ there exists a number between $n^2$ and $(n+1)^2$ with at most 4 prime factors. This is the first result of this kind that holds for every $n\geq 1$ rather than just sufficiently large $n$. Our approach…

Let $X$ be large and let $\mathcal{P}$ denote the set of primes. Fix positive real parameters $r_1,\dots,r_s$ and a parameter $\lambda\geqslant 1$ determined by a balancing relation, and let $\mathcal{A}_{\lambda}(X)\subset[1,2X]$ be the…