On asymptotic behaviour of Dirichlet inverse

Abstract

Let $f(n)$ be an arithmetic function with $f(1)\ne 0$ 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.

16 pages. Section 3.1 has been slightly expanded. Minor improvements have been incorporated according to referee's suggestions. Accepted to International Journal of Number Theory