Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative

Pentti Haukkanen, Jorma K. Merikoski, Timo Tossavainen


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 $$\sum_{1\le n\le x}\frac{n'}{n^s},\,\sum_{1\le n\le x}\frac{n'_p}{n^s},\quad{\rm and}\,\,\sum_{1\le n\le x}\frac{n'_P}{n^s}.$$ We also show that the abscissa of convergence of the corresponding Dirichlet series equals~two.


Abscissa of convergence; arithmetic derivative; Dirichlet series

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ISSN: 1331-0623 (Print), 1848-8013 (Online)