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 \( -\sum _{\begin{array}{c} n\ge 2\\ P^-(n)\equiv \ell (\textrm{mod}k) \end{array}}\frac{\mu (n)}{n}=\frac{1}{\varphi (k)} \) for positive integers \(\ell , k\ge 1\) 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.
