  In this article, we prove that a general version of Alladi's formula with
Dirichlet convolution holds for arithmetical semigroups satisfying Axiom $A$ or
Axiom $A^{\#}$. As applications, we apply our main results to certain
semigroups coming from algebraic number theory, arithmetical geometry and graph
theory, particularly generalizing the results of Wang 2021, Kural et al. 2020
and Duan et al. 2020.

Number Theory