Related papers: Gaps of Smallest Possible Order between Primes in …

Let $\{Z_t, t\geq 0\}$ be a strictly stable process on $\R$ with index $\alpha\in (0,2]$. We prove that for every $p > \alpha$, there exists $\gamma = \gamma (\alpha, p)$ and $\k = \k (\alpha, p)\in (0, +\infty)$ such that…

We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if $t$ is an integer different from $0, 1$ or -1 and if $\A \subset \Zp$ is not too large (with respect to $p$),…

The following is proven using arguments that do not revolve around the Riemann Hypothesis or Sieve Theory. If $p_n$ is the $n^{\rm th}$ prime and $g_n=p_{n+1}-p_n$, then $g_n=O({p_n}^{2/3})$.

Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a nonempty subset of $\mathbb{F}_p$. In this paper we show that if $|A|\preceq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succeq|A|^{13/12};\] if…

Let $\Psi$ be a system of linear forms with finite complexity. In their seminal paper, Green and Tao showed the following prime number theorem for values of the system $\Psi$: $$\sum_{x\in [-N,N]^d} \prod_{i=1}^t…

A natural number $n$ is called semi-prime if it is a product of two primes or a square of a prime. We denote $\mathbb{P}_2$ the set of all semi-primes. Our goal is to prove that for fixed integer number $a$ and sufficiently large $x$ the…

Let $1<a<b$ be two relatively prime integers and $\mathbb{Z}_{\ge 0}$ the set of non-negative integers. For any non-negative integer $\ell$, denote by $g_{\ell,a,b}$ the largest integer $n$ such that the equation $$n=ax+by,\quad…

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…

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…

For a prime number $p$ and a sequence of integers $a_0,\dots,a_k\in \{0,1,\dots,p\}$, let $s(a_0,\dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,\dots,x_k)\in A_0\times\dots\times A_k$ with $x_0=x_1+\dots + x_k$, over subsets…

For real $\xi$ we consider the irrationality measure function $\psi_\xi(t) = \min_{1\leqslant q \leqslant t, q\in\mathbb{Z}} || q\xi ||$, where $||\cdot||$ - distance to the nearest integer. We prove that in the case…

Let $a>1$ be an integer. Denote by $l_a(p)$ the multiplicative order of $a$ modulo primes $p$. We prove that if $\frac{x}{\log x\log\log x}=o(y)$, then $$\frac 1 y \sum_{a\leq y}\sum_{p\leq x}\frac{1}{l_a(p)}=\log x + C\log\log…

We show that there exists some $\delta > 0$ such that, for any set of integers $B$ with $B\cap[1,Y]\gg Y^{1-\delta}$ for all $Y \gg 1$, there are infinitely many primes of the form $a^2+b^2$ with $b\in B$. We prove a quasi-explicit formula…

Suppose that $1<c<9/8$. For any $m\geq 1$, there exist infinitely many $n$ such that $$ \{[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]\} $$ contains at least $m+1$ primes, if $k_0$ is sufficiently large (only depending on $m$).

Let $f(x_1,\ldots,x_s)$ be a translation invariant indefinite quadratic form of integer coefficients with $s\ge 10$. Let $\mathcal{A}\subseteq \mathcal{P}\cap \{1,2,\ldots,X\}$. Let $X$ be sufficiently large. Subject to a rank condition, we…

In this paper, if prime $p\equiv 3\pmod 4$ is sufficiently large then we prove an upper bound on the number of occurences of any arbitrary pattern of quadratic residues and nonresidues of length $k$ as $k$ tends to $\lceil \log_2 p\rceil$.…

Let $\mathcal{G}$ be the greedy algorithm that, for each $\theta\in (0,1]$, produces an infinite sequence of positive integers $(a_n)_{n=1}^\infty$ satisfying $\sum_{n=1}^\infty 1/a_n = \theta$. For natural numbers $p < q$, let…

Let $q_1, \ldots , q_t$ be distinct prime numbers. Let $a_1, \ldots , a_t$ be nonnegative integers. We establish effective lower bounds for $|z^d - q_1^{a_1} \ldots q_t^{a_t}|$ and for its greatest prime factor, which tend to infinity with…

95 years ago Hoheisel proved the existence of primes in the sub-linear interval \[ \left[x, x+x^{1-{1\over 33000}}\right] \qquad \hbox{for $x$ sufficiently large}. \] This was improved by Heilbronn, proving existence of primes in the…

The author proves that there are infinitely many primes $p$ such that $\| \alpha p - \beta \| < p^{-\frac{28}{87}}$, where $\alpha$ is an irrational number and $\beta$ is a real number. This sharpens a result of Jia (2000) and provides a…