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

In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+x^epsilon] is about x^epsilon/log x and the second says that the number of primes…

Let $Q(n)$ denote the number of integers $1 \leq q \leq n$ whose prime factorization $q= \prod^{t}_{i=1}p^{a_i}_i$ satisfies $a_1\geq a_2\geq \ldots \geq a_t$. Hardy and Ramanujan proved that $$ \log Q(n) \sim \frac{2\pi}{\sqrt{3}}…

Let $f : \mathbf{N} \rightarrow \mathbf{C}$ be a bounded multiplicative function. Let $a$ be a fixed integer (say $a = 1$). Then $f$ is well-distributed on the progression $n \equiv a \pmod{q} \subset \{1,\dots, X\}$, for almost all primes…

Let $P(a,q)$ be the least prime in the arithmetic progression $\{n\equiv a(mod\ q)\}$. In this note, when $q$ has bounded cubic part and $(a,q)=1$, we combine the Heath-Brown's method and the Burgess's bounds for L-functions to obtain $…

Let p1, p2,..., pn be distinct prime numbers, and let Nn be their product. We prove that, for any positive integer L that is divisible by the least common multiple of p1 minus one, p2 minus one, and so on, and for integers a1, a2,..., an…

We prove several unique prime factorization results for tensor products of type II_1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected Lie groups of real rank 1.…

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…

We show that for any irrational $\alpha$ and any $\tau<8/23$ there are infinitely many $n$ which are the product of two primes for which $$\|n\alpha\|\leq n^{-\tau}.$$ We also show that for all sufficiently large $b$ there exist 3-digit…

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical…

Let $\omega(n)$ denote the number of distinct prime factors of $n$. Then for any given $K\geq 2$, small $\epsilon>0$ and sufficiently large (only depending on $K$ and $\epsilon$) $x$, there exist at least $x^{1-\epsilon}$ integers…

For $(M,a)=1$, put \begin{equation*} G(X;M,a)=\sup_{p^\prime_n\leq X}(p^\prime_{n+1}-p^\prime_n), \end{equation*} where $p^\prime_n$ denotes the $n$-th prime that is congruent to $a\pmod{M}$. We show that for any positive $C$, provided $X$…

Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a…

Let $L_1$, $L_2$ $L_3$ be integer linear functions with no fixed prime divisor. We show there are infinitely many $n$ for which the product $L_1(n)L_2(n)L_3(n)$ has at most 7 prime factors, improving a result of Porter. We do this by means…

We show that for any $k$-times continuously differentiable function $f:[a,\infty)\longrightarrow{\mathbb R}$, any integer $q\ge 0$ and any $\alpha>1$ the inequality $$\liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\dots\cdot…

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for $\alpha\in\mathbb{R}\backslash\mathbb{Q},\,\beta\in\mathbb{R}$ and $0<\theta<10/1561$, there…

We prove, in particular, the well--known Zaremba conjecture from the theory of continued fractions for any prime denominator. More precisely, we show, firstly, that under some mild conditions, for any sufficiently large $q$, there exists…

Given an integer $m \geq 2$ and a sufficiently large $q$, we apply a variant of the Maynard--Tao sieve weight to establish the existence of an arithmetic progression with common difference $q$ for which the $m$-th least prime in such…

We show that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p,q$ both prime, satisfy $p\equiv q\equiv 3 \pmod 4$.

Let $\pi_{q,a}(x)$ denote the number of primes $\le x$ in the progression $a$ modulo $q$. We study subtle inequities in these functions, with $q$ fixed and variable $a$ (sometimes called 'prime race problems'). It is known unconditionally…

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…