arXiv subjects

Mihoub Bouderbala

Publications and source records attributed to Mihoub Bouderbala.

5 recordsLinked to original sources

arXiv2026

On the Mean Value of a Weighted Composite Arithmetic Function

The primary objective of this paper is to employ methods from analytic number theory to investigate the mean value properties of a composite function involving the Dirichlet divisor function and a generalized minimal power function. Specifically, we study the weighted summatory function where the divisor function is normalized by the number of distinct prime factors. We establish a rigorous asymptotic formula for this sum, detailing the analytic properties of the associated Dirichlet series and the contour integration process.

math.NT↗

arXiv2026

On certain sums involving the largest prime factor over integer sequences

Given an integer $n \ge 2$, its prime factorization is expressed as $n= \prod_{i=1}^s p_i^{a_i}$. We define the function $f(n)$ as the smallest positive integer such that $f(n)!$ is divisible by $n$. The main objective of this paper is to derive an asymptotic formula for both sums $\sum_{n \le x} f(n)$ and $\sum_{n \le x, n \in S_k} f(n)$, where $S_k$ denotes the set of all $k$-free integers.

math.NT↗

arXiv2026

On the asymptotic behavior of the integral T(x)

In this paper, we estimate the integral T(x) mentioned in the title, where {t} denotes the fractional part of the real number t, and x is any positive real number.

math.NT↗

arXiv2022

On a sum of a multiplicative function linked to the divisor function over the set of integers B-multiple of 5

Let $d(n)$ and $d^{\ast}(n)$ be the numbers of divisors and the numbers of unitary divisors of the integer $n\geq1$. In this paper, we prove that \[ \underset{n\in\mathcal{B}}{\underset{n\leq x}{\sum}}\frac{d(n)}{d^{\ast}% (n)}=\frac{16π% %TCIMACRO{\U{b2}}% %BeginExpansion {{}^2}% %EndExpansion }{123}\underset{p}{\prod}(1-\frac{1}{2p% %TCIMACRO{\U{b2}}% %BeginExpansion {{}^2}% %EndExpansion }+\frac{1}{2p^{3}})x+\mathcal{O}\left( x^{\frac{\ln8}{\ln10}+\varepsilon }\right) ,~\left( x\geqslant1,~\varepsilon>0\right) , \] where $\mathcal{B}$ is the set which contains any integer that is not a multiple of $5,$ but some permutations of its digits is a multiple of $5.$

math.NT↗

arXiv2019

Sur la variation de certaines suites de parties fractionnaires

Let $b > a > 0$. We prove the following asymptotic formula $$\sum_{n\ge 0} \big\lvert\{x/(n+a)\} - \{x/(n+b)\}\big\rvert = \frac{2}πζ(3/2)\sqrt{cx} + O(c^{2/9}x^{4/9}),$$ with $c=b-a$, uniformly for $x \ge 40 c^{-5}(1+b)^{27/2}$.

math.NT↗