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

Fix a modulus $q$. One would expect the number of primes in each invertible residue class mod $q$ to be multinomially distributed, i.e. for each $p \,\mathrm{mod}\, q$ to behave like an independent random variable uniform on…

We provide a formula for the logarithmic density of the set of positive real numbers on which two prime counting functions $\psi(x;q,a)$ and $\psi(x;q,b)$ are simultaneously larger than their asymptotic main terms, as well as a method for…

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…

We show that for every fixed $A>0$ and $\theta>0$ there is a $\vartheta=\vartheta(A,\theta)>0$ with the following property. Let $n$ be odd and sufficiently large, and let $Q_{1}=Q_{2}:=n^{\h}(\log n)^{-\vartheta}$ and $Q_{3}:=(\log…

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\varphi(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12,…

In the first paper under this title (1977), the first author utilized a duality identity between the largest and smallest prime factors involving the Moebius function, to establish the following result as a consequence of the Prime Number…

Let $l$ and $m$ be two integers with $l>m\ge 0$, and let $f(x)$ be the product of two linear polynomials with integer coefficients. In this paper, we show that $\log {\rm lcm}_{mn<i\le ln}\{f(i)\}=An+o(n)$, where $A$ is a constant depending…

We prove that there are infinitely many $n$ such that $\omega(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao-Ter\"{a}v\"{a}inen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress…

Let $\gamma_{a,b}(n)$ be the number of smooth words of length $n$ over the alphabet $\{a,b\}$ with $a<b$. Say that a smooth word $w$ is \emph{left fully extendable} (LFE) if both $aw$ and $bw$ are smooth. In this paper, we prove that for…

We obtain a totient analogue for Linnik's theorem in arithmetic progressions. Specifically, for any coprime pair of positive integers $(m,a)$ such that $m$ is odd, there exists $n\le m^{2+o(1)}$ such that $\varphi(n)\equiv…

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…

We show that if $A\subset \mathbb{Z}$ is a finite set of integers in which every integer is divisible by $O(1)$ many primes then \[\max(\lvert A+A\rvert,\lvert AA\rvert) \geq \lvert A\rvert^{12/7-o(1)}\] and, for any $m\geq 2$,…

It is well known that a polynomial $\phi(X)\in \mathbb{Z}[X]$ of given degree $d$ factors into at most $d$ factors in $\mathbb{F}_p$ for any prime $p$. We prove in this paper the existence of infinitely many primes $q$ so that the given…

We give several families of reasonably small integers $k, \ell \ge 1$ and real positive $\alpha, \beta \le 1$, such that the products $p_1\ldots p_k s$, where $p_1, \ldots, p_k \le m^\alpha$ are primes and $s \le m^\beta$ is a product of at…

Let \beta be a real number. Then for almost all irrational \alpha>0 (in the sense of Lebesgue measure) \limsup_{x\to\infty}\pi_{\alpha,\beta}^*(x)(\log x)^2/x>=1, where \pi_{\alpha,\beta}^*(x)={p<=x: both p and [\alpha p+\beta] are primes}.

For any real $k\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum \begin{equation*} \frac{1}{\phi(q)} \sum_{\substack{\chi \text{ mod }q\\ \chi\neq \chi_0}} \Big| \sum_{n\leq x}…

Let $\|x\|$ denote the distance from $x\in\mathbb{R}$ to the nearest integer. In this paper, we prove an existence and density statement for matrices $\boldsymbol{A}\in\mathbb{R}^{m\times n}$ satisfying…

Let $k$ be a positive real number, and let $M_k(q)$ be the sum of $|L(\tfrac12,\chi)|^{2k}$ over all non-principal characters to a given modulus $q$. We prove that $M_k(q)\ll_k \phi(q)(\log q)^{k^2}$ whenever $k$ is the reciprocal $n^{-1}$…

In this paper, we investigate the quantity $m_q:=\min_{\chi\ne \chi_0} | L^\prime/L(1,\chi)|$, as $q\to \infty$ over the primes, where $L(s,\chi)$ is the Dirichlet $L$-function attached to a non trivial Dirichlet character modulo $q$. Our…

Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek lower bounds for $P(k)$. In particular,…