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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
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