The mean value of the function \frac{d(n)}{d^*(n)} in arithmetic progressions

Abstract

Let $d(n)$ and $d^*(n)$ be, respectively, the number of divisors and the number of unitary divisors of an integer $n\geq 1.$ A divisor $d$ of an integer is to be said unitary if it is prime over $\frac{n}{d}.$ In this paper, we study the mean value of the function $D(n)=\frac{d(n)}{d^*(n)}$ in the arithmetic progressions $ \left\lbrace l+mk \mid m\in\mathbb{N}^* \text{ and } (l, k)=1 \right\rbrace;$ this leads back to the study of the real function $x\mapsto S(x;k,l)=\underset{n\equiv l[k]}{\sum\limits_{ n \leq x}} D(n).$ We prove that $$ S(x;k,l)=A(k)x +\mathcal{O}_{k}\left(x\exp \left( -\frac{\theta}{2}\sqrt{(2\ln x)(\ln\ln x)}\right) \right) \left( 0<\theta<1 \right),$$ where $\quad A(k)=\dfrac{c}{k}\prod\limits_{p\mid k}\left(1+\dfrac{1}{2}\sum\limits_{n=2}^{+\infty}\dfrac{1}{p^{n}}\right)^{-1}\left( c=\zeta(2)\prod\limits_{p} \left(1-\dfrac{1}{2p^2}+\dfrac{1}{2p^3} \right) \right).$