Analogues of Alladi's formula

Abstract: In this note, we mainly show the analogue of one of Alladi's formulas over $\mathbb{Q}$ with respect to the Dirichlet convolutions involving the M\"{o}bius function $\mu(n)$, which is related to the natural densities of sets of primes by recent work of Dawsey, Sweeting and Woo, and Kural et al. This would give us several new analogues. In particular, we get that if $(k, \ell)=1$, then $$-\sum_{\begin{smallmatrix}n\geq 2\ p(n)\equiv \ell (\operatorname{mod} k) \end{smallmatrix}} \frac{\mu(n)}{\varphi(n)} = \frac1{\varphi(k)},$$ where $p(n)$ is the smallest prime divisor of $n$, and $\varphi(n)$ is Euler's totient function. This refines one of Hardy's formulas in 1921. At the end, we give some examples for the $\varphi(n)$ replaced by functions "near $n$", which include the sum-of-divisors function.