Related papers: Variance of primes in short residue classes for fu…

We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms,…

The main objective of this article is to study the asymptotic behavior of Salie sums over arithmetic progressions. We deduce from our asymptotic formula that Salie sums possess a bias of being positive. The method we use is based on…

On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

In this article, we prove an "equivalence" between two higher even moments of primes in short intervals under Riemann Hypothesis. We also provide numerical evidence in support of these asymptotic formulas.

Statistical distribution of the primes in an arithmetic progression is considered. The estimation of prime numbers is given and combinatorial methods are used to calculate the twin primes on the available interval. The distribution and…

We prove an asymptotic formula for the number of primes of the shape $a^2 +p^4$, thereby refining the well known work of Friedlander and Iwaniec. Along the way, we prove a result on equidistribution of primes up to $x$, in which the moduli…

We give asymptotics for correlation sums linked with the distribution of squarefree numbers in arithmetic progressions over a fixed modulus. As a particular case we improve a result of Blomer concerning the variance.

Primitive polynomials over finite fields are crucial for various domains of computer science, including classical pseudo-random number generation, coding theory and post-quantum cryptography. Nevertheless, the pursuit of an efficient…

We prove an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields.

The work is devoted to the construction of a new interval arithmetic which would combine algorithmic efficiency and high quality estimation of the ranges of expressions.

In this work we obtain recurrent formulae for the number of permutations with either increasing or monotonic (i.e., both increasing and decreasing) runs of bounded length. Our formulae allow one to efficiently compute the number of such…

We investigate, using the weighted linear sieve, the distribution of almost-primes among the residue classes (mod p) that generate the multiplicative group of reduced residue classes. We are concerned with finding an upper bound for the…

In the present paper we prove that there exist infinitely many arithmetic progressions of three different primes $p_1,p_2,p_3=2p_2-p_1$ such that $p_1=x_1^2 + y_1^2 +1$, $p_2=x_2^2 + y_2^2 +1$.

Given good knowledge on the even moments, we derive asymptotic formulas for $\lambda$-th moments of primes in short intervals and prove "equivalence" result on odd moments. We also provide numerical evidence in support of these results.

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…

Interval linear programming provides a tool for solving real-world optimization problems under interval-valued uncertainty. Instead of approximating or estimating crisp input data, the coefficients of an interval program may perturb…

In the present paper, we adopt a pretentious approach and prove a strongly uniform estimate for the sums of the von Mangoldt function $\Lambda$ on arithmetic progressions. This estimate is analogous to an estimate that Linnik established in…

In this paper we establish the function field analogue of Bateman-Horn conjecture in short interval in the limit of a large finite field. Hence we start with counting prime polynomials generated by primitive quadratic functions in short…

We obtain upper bounds on the number of solutions to congruences of the type $$ (x_1+s)...(x_{\nu}+s)\equiv (y_1+s)...(y_{\nu}+s)\not\equiv0 \pmod p $$ modulo a prime $p$ with variables from some short intervals. We give some applications…

Let $q\ge 3$ be a non-exceptional modulus $q\ge3$, and let $a$ be a positive integer coprime with $q$. For any $\epsilon>0$, there exists $\alpha>0$ (computable), such that for all $x\ge \alpha (\log q)^2$, the interval $\left[…