Analogues of Alladi's formula

In this note, we mainly show the analogue of one Alladi's formulas over Q with respect to Dirichlet convolutions involving Möbius function μ(n), which is related natural densities sets primes by recent work Dawsey, Sweeting and Woo, Kural et al. This would give us several new analogues. particular, get that if (k,ℓ)=1, then−∑n⩾2p(n)≡ℓ(modk)μ(n)φ(n)=1φ(k), where p(n) smallest prime divisor n, φ(n) Euler's totient function. refines Hardy's in 1921. At end, some examples for replaced functions “near n”, include sum-of-divisors