Geodesic
On some integrals involving $\Delta_2(x)$ and $\Delta_3(x)$
Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 46 (2021) no. 1 
Wenguang Zhai. On some integrals involving $\Delta_2(x)$ and $\Delta_3(x)$. Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 46 (2021) no. 1. http://geodesic.mathdoc.fr/item/BASS_2021_46_1_a5/
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     title = {On some integrals involving $\Delta_2(x)$ and $\Delta_3(x)$},
     journal = {Bulletin de l'Acad\'emie serbe des sciences. Classe des sciences math\'ematiques et naturelles},
     pages = {93 - 114},
     year = {2021},
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     number = {1},
     url = {http://geodesic.mathdoc.fr/item/BASS_2021_46_1_a5/}
}
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Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

Let $k\geq 2$ be a fixed natural number and $d_k(n)$ denote the number of ways $n$ can be written as a product of $k$ positive integers. Let $\Delta_k(x)$ denote the error term in the asymptotic formula of the summatory function of $d_k(n)$. In this paper we study some integrals involving $\Delta_2(x)$ and $\Delta_3(x)$.