Sums of quotients of additive functions

Abstract:

Denote by $\omega (n)$ and $\Omega (n)$ the number of distinct prime factors of $n$ and the total number of prime factors of $n$, respectively. Given any positive integer $\alpha$, we prove that \[ \sum \limits _{2 \leqq n \leqq x} {\Omega (n)/\omega } (n) = x + x\sum \limits _{i = 1}^\alpha {{a_i}/{{(\log \log x)}^i} + O{{(x/\log \log x)}^{\alpha + 1}}),} \] where ${a_1} = \sum \nolimits _p {1/p(p - 1)}$ and all the other ${a_i}$’s are computable constants. This improves a previous result of R. L. Duncan.