an:07200858
Zbl 07200858
Haukkanen, Pentti; Merikoski, Jorma K.; Tossavainen, Timo
Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative
EN
Math. Commun. 25, No. 1, 107-115 (2020).
00449225
2020
j
11N37 11N56
abscissa of convergence; arithmetic derivative; Dirichlet series
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.