×

zbMATH — the first resource for mathematics

Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative. (English) Zbl 07200858
Summary: Let \(p\in\mathbb{P}\) and \(s\in\mathbb{R}\), and suppose that \(\emptyset\ne P\subset\mathbb{P}\) is finite. Given \(n\in\mathbb{Z}_+\), let \(n'\), \(n'_p\), and \(n'_P\) denote respectively its arithmetic derivative, arithmetic partial derivative with respect to \(p\), and arithmetic subderivative with respect to \(P\). We study the asymptotics of \[\sum_{1\le n\le x}\frac{n'}{n^s},\,\sum_{1\le n\le x}\frac{n'_p}{n^s},\text{ and } \sum_{1\le n\le x}\frac{n'_P}{n^s}.\] We also show that the abscissa of convergence of the corresponding Dirichlet series equals two.

MSC:
11N37 Asymptotic results on arithmetic functions
11N56 Rate of growth of arithmetic functions
PDF BibTeX XML Cite
Full Text: Link