Deep Thakur

On a series related to the M\"obius function

For a natural number $n>1$ let $P^-(n)$ denote its smallest prime divisor. Alladi and Johnson proved that $\sum_{P^{-}(n)\equiv l \Mod{k}} \frac{\mu(n)\omega(n)}{n} = 0$ 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)...