# Asymptotic Expansion of Warlimont Functions on Wright Semigroups

## TL;DR

This paper derives detailed asymptotic expansions for multiplicative functions on Wright semigroups, providing insights into graph decomposition and polynomial structures over finite fields.

## Contribution

It introduces a method to compute full asymptotic expansions of prime-independent multiplicative functions on Wright semigroups, extending Knopfmacher's axioms.

## Key findings

- Asymptotic expansions for graph decomposition into connected components
- Application to polynomials over finite fields
- Enhanced understanding of additive arithmetic semigroups

## Abstract

We calculate full asymptotic expansions of prime-independent multiplicative functions on additive arithmetic semigroups that satisfy a strong form of Knopfmacher's axioms. When applied to the semigroup of unlabeled graphs, our method yields detailed asymptotic information on how graphs decompose into connected components. As a second class of examples, we discuss polynomials in several variables over a finite field.