Related papers: Inequities in the Shanks-Renyi prime number race o…

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let $\ell$ be a rational prime and $K$ a rational function field $\Bbb F_q(t)$ with $\ell \nmid q$.…

We define a new metric between natural numbers induced by the $\ell_\infty$ norm of their unique prime signatures. In this space, we look at the natural analog of the number line and study the arithmetic function $L_\infty(N)$, which…

We prove a function field analogue of Maynard's result about primes with restricted digits. That is, for certain ranges of parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a…

In this paper, we introduce the prime trees associated with a finite subset $P$ of the set of all prime numbers, and provide conditions under which the tree is of finite type. Moreover, we compute the density of finite-type subsets $P$. As…

This note investigates the average density of prime numbers $p\in[x,2x]$ with respect to a random simultaneous primitive root $g\leq p^{1/2+\varepsilon}$ over the finite rings $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}/p^2\mathbb{Z}$ as $x…

We develop an analytic approach that draws on tools from Fourier analysis and ergodic theory to study Ramsey-type problems involving sums and products in the integers. Suppose $Q$ denotes a polynomial with integer coefficients. We establish…

Let $\mathbb{F}_q$ denote the finite field of $q$ elements and $\mathbb{F}_{q^n}$ the degree $n$ extension of $\mathbb{F}_q$. A normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F} _q$ is a basis of the form…

We compute the leading asymptotics as $N\to\infty$ of the maximum of the field $Q_N(q)= \log\det|q- A_N|$, $q\in \mathbb{C}$, for any unitarily invariant Hermitian random matrix $A_N$ associated to a non-critical real-analytic potential.…

Let $F$ be an integral linear recurrence, $G$ be an integer-valued polynomial splitting over the rationals, and $h$ be a positive integer. Also, let $\mathcal{A}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd(F(n), G(n)) =…

Let $a_1 = 1$ and, for $n > 1$, $a_n = a_{n-1} + a_{\left \lfloor \frac{n}{2} \right \rfloor}$. In this paper we will look at congruence properties and the growth rate of this sequence. First we will show that if $x \in \{1, 2, 3, 5, 6, 7…

Consider a fair lottery over the natural numbers in which the selected number is removed. This lottery is iterated countably infinite times, with a known ratio of iterations to natural numbers. Removed numbers are not replaced. The natural…

We obtain new uniform bounds for the symmetric tensor rank of multiplication in finite extensions of any finite field Fp or Fp2 where p denotes a prime number greater or equal than 5. In this aim, we use the symmetric Chudnovsky-type…

We examine the behavior of the coefficients of powers of polynomials over a finite field of prime order. Extending the work of Allouche-Berthe, 1997, we study a(n), the number of occurring strings of length n among coefficients of any power…

Let $n_1,\cdots,n_r$ be any finite sequence of integers and let $S$ be the set of all natural numbers $n$ for which there exists a divisor $d(x)=1+\sum_{i=1}^{deg(d)}c_ix^i$ of $x^n-1$ such that $c_i=n_i$ for $1\leq i \leq r$. In this paper…

All the known approximations of the number of primes pi(n) not exceeding any given integer n are derived from real-valued functions that are asymptotic to pi(x), such as x/log x, Li(x) and Riemann's function R(x). The degree of…

Let $F_1,\ldots,F_R$ be homogeneous polynomials with integer coefficients in $n$ variables with differing degrees. Write $\boldsymbol{F}=(F_1,\ldots,F_R)$ with $D$ being the maximal degree. Suppose that $\boldsymbol{F}$ is a nonsingular…

Let P be the set of all primes and P_2=P\cup{p_1p_2: p_1,p_2\in P}$. We prove that the sumset 2^P+P_2={2^p+q: p\in P, q\in P_2} has a positive lower density.

In a recent paper, Bary-Soroker, Koukoulopoulos and Kozma proved that when $A$ is a random monic polynomial of $\mathbb{Z}[X]$ of deterministic degree $n$ with coefficients $a_j$ drawn independently according to measures $\mu_j,$ then $A$…

We study the number of prime polynomials of degree $n$ over $\mathbb{F}_q$ in which the $i^{th}$ coefficient is either preassigned to be $a_i \in \mathbb{F}_q$ or outside a small set $S_i \subset \mathbb{F}_q$. This serves as a function…

We define a necessary and sufficient condition on a polynomial $h\in \mathbb{Z}[x]$ to guarantee that every set of natural numbers of positive upper density contains a nonzero difference of the form $h(p)$ for some prime $p$. Moreover, we…