On a series related to the M\"obius function

Abstract

For a natural number $n>1$ let $P^-(n)$ denote its smallest prime divisor. Alladi and Johnson proved that $\sum_{P^{-}(n)\equiv l \Mod{k}} \frac{\mu(n)\omega(n)}{n} = 0$ for $(l,k)=1$, and Tenenbaum showed that the partial sums of this series up to $X$ are $O_\varepsilon\bigl((\log\log X)^{5/3+\varepsilon}/\log X\bigr)$. In this article, we prove that these partial sums are in fact equal to $C(k,l)/\log X + O\bigl((\log\log X)^4/(\log X)^2\bigr)$, 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 $l\Mod{k}$ for which $C(k,l)<0$ is $\varphi(k)(1- o(1))$ as $k\to\infty$.