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 ${\sum }_{P^{-}(n)\equiv l\text{Mod}k}\frac{\mu (n)\omega (n)}{n}=0$ for $(l,k)=1$, and Tenenbaum showed that the partial sums of this series up to $X$ are $O_{\epsilon }((\log \log X{)}^{5\mathrm{/}3+\epsilon }\mathrm{/}\log X)$. In this article, we prove that these partial sums are in fact equal to $C(k,l)\mathrm{/}\log X+O((\log \log X{)}^4\mathrm{/}(\log X{)}^2)$, with an explicit constant $C(k,l)$. We also investigate the behaviour of these constants. In particular, we show under GRH that the number of reduced classes $l\text{Mod}k$ for which $C(k,l)<0$ is $\phi (k)(1-o(1))$ as $k\to \mathrm{\infty }$.