Generalizations of Alladi's formula for arithmetical semigroups

TL;DR

This paper extends Alladi's formula with Dirichlet convolution to a broad class of arithmetical semigroups, providing new generalizations and applications across number theory, geometry, and graph theory.

Contribution

It introduces a generalized Alladi's formula applicable to arithmetical semigroups satisfying Axiom A or Axiom A#, expanding previous results to new algebraic and combinatorial contexts.

Findings

Generalized Alladi's formula for semigroups satisfying Axiom A or A#

Applications to algebraic number theory, arithmetical geometry, and graph theory

Extension of previous specific results to broader classes of 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 Wang 2021, Kural et al. 2020 and Duan et al. 2020.