# On asymptotic behaviour of Dirichlet inverse

## TL;DR

This paper investigates the asymptotic properties of the Dirichlet inverse of an arithmetic function, providing bounds and insights based on the growth or decay rates of the original function.

## Contribution

It offers new asymptotic estimates for the Dirichlet inverse and introduces bounds for counting factorizations of integers into multiple factors.

## Key findings

- Derived bounds for |f^{-1}(n)| based on |f(n)| growth rates
- Established upper bounds for the number of factorizations of n into k factors
- Connected asymptotic behaviors of functions and their Dirichlet inverses

## Abstract

Let $f(n)$ be an arithmetic function with $f(1)\neq0$ and let $f^{-1}(n)$ be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behaviour of $|f^{-1}(n)|$ with regard to the asymptotic behaviour of $|f(n)|$ assuming that the latter one grows or decays with at most polynomial or exponential speed. As a by-product, we obtain simple but constructive upper bounds for the number of ordered factorizations of $n$ into $k$ factors.