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

TL;DR

This paper derives an asymptotic formula for the sum of the ratio of divisor counts over a special set of integers related to digit permutations, revealing new connections between divisor functions and digit-based sets.

Contribution

It introduces a novel analysis of divisor functions over a set of integers defined by digit permutation properties, providing an explicit asymptotic formula.

Findings

Asymptotic formula for the sum involving divisor functions over the set -multiple of 5.

Explicit constant involving an infinite product over primes.

Error term characterized by a specific exponent.

Abstract

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 $5.$