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On asymptotic behavior of Dirichlet inverse

Department of Mathematics, University of Rostock, Ulmenstraße 69, Rostock 18057, Germany

and

Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, Plzeň 301 00, Czech Republic

Institute of Mathematics, Ufa Federal Research Centre, RAS, Chernyshevsky str. 112, Ufa 450008, Russia

E-mail Address: [email protected]

Corresponding author.

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https://doi.org/10.1142/S1793042120500700Cited by:1 (Source: Crossref)

Abstract

Let $f (n)$ be an arithmetic function with $f (1) \neq 0$ and let $f^{- 1} (n)$ be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behavior of $| f^{- 1} (n) |$ with regard to the asymptotic behavior 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.

Keywords:

AMSC: 11A25, 11N37, 11N56, 05A16, 05A17

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