A logarithmic analogue of Alladi's formula

Abstract: Let $\mu(n)$ be the M\"{o}bius 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_{\substack{n\geq 2\ P^-(n)\equiv \ell ({\rm mod}k)}}\frac{\mu(n)}{n}=\frac1{\varphi(k)} ] for positive integers $\ell, k\ge$ with $(\ell,k)=1$, where $\varphi$ is Euler's totient function. In this note, we will show a logarithmic analogue of Alladi's formula in an elementary proof.