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

cm:10467 - Communications in Mathematics, December 22, 2022, Volume 31 (2023), Issue 1 - https://doi.org/10.46298/cm.10467
On a sum of a multiplicative function linked to the divisor function over the set of integers B-multiple of 5

Authors: Mihoub Bouderbala

    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 \[ \underset{n\in\mathcal{B}}{\underset{n\leq x}{\sum}}\frac{d(n)}{d^{\ast}% (n)}=\frac{16\pi% %TCIMACRO{\U{b2}}% %BeginExpansion {{}^2}% %EndExpansion }{123}\underset{p}{\prod}(1-\frac{1}{2p% %TCIMACRO{\U{b2}}% %BeginExpansion {{}^2}% %EndExpansion }+\frac{1}{2p^{3}})x+\mathcal{O}\left( x^{\frac{\ln8}{\ln10}+\varepsilon }\right) ,~\left( x\geqslant1,~\varepsilon>0\right) , \] 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 $5.$


    Volume: Volume 31 (2023), Issue 1
    Published on: December 22, 2022
    Accepted on: December 13, 2022
    Submitted on: December 13, 2022
    Keywords: Mathematics - Number Theory,11A25, 11N37

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