On a series related to the Möbius function

For a natural number $n>1$ let $P^-(n)$ denote its smallest prime divisor. Alladi and Johnson proved that $\sum_{P^{-}(n)\equiv l \Mod{k}} \frac{μ(n)ω(n)}{n} = 0$ for $(l,k)=1$, and Tenenbaum showed that the partial sums of this series up to $X$ are $O_\varepsilon\bigl((\log\log X)^{5/3+\varepsilon}