A logarithmic analogue of Alladi's formula

Let $\mu (n)$ be the Möbius function. Let $P^{-}(n)$ denote the smallest prime factor of an integer $n$. In 1977, Alladi established the following formula related to the prime number theorem for arithmetic progressions $$-\sum _{\begin{array}{c}n\ge 2\\ P^{-}(n)\equiv \mathrm{\ell }(\mathrm{m}\mathrm{o}\mathrm{d}k)\end{array}}\frac{\mu (n)}{n}=\frac{1}{\phi (k)}$$ for positive integers $\mathrm{\ell },k\ge$ with $(\mathrm{\ell },k)=1$, where $\phi$ is Euler’s totient function. In this note, we will show a logarithmic analogue of Alladi’s formula in an elementary proof.