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On asymptotic behavior of Dirichlet inverse. (English) Zbl 1457.11010

Let \(f\) be an arithmetic function with \(f(1)\ne 0\), and let \(f^{-1}\) denote its Dirichlet inverse. The authors find upper bounds for \(\vert f^{-1}(n)\vert\) assuming that \(\vert f(n)\vert\) grows or decays with at most polynomial or exponential speed, that is, \(\vert f(n)\vert\le Cn^{\gamma}\) or \(\vert f(n)\vert\le Ac^{n}\) for some \(C>0\), \(\gamma\in\mathbb{R}\) and \(A, c>0\).
The examination is divided into multiplicative functions and the general case. As a by-product, upper bounds for the number of ordered factorizations of \(n\) into \(k\) factors are obtained.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11N37 Asymptotic results on arithmetic functions
11N56 Rate of growth of arithmetic functions
05A16 Asymptotic enumeration
05A17 Combinatorial aspects of partitions of integers
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References:

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