Generalizations of Alladi's formula for arithmetical semigroups

Abstract

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 [27,17,8].