Sums of quotients of additive functions

Abstract

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