Some remarks on a paper of L. Tóth
Journal of integer sequences, Tome 13 (2010) no. 1.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Consider the functions $P(n):=\sum_{k=1}^n \gcd(k,n)$ (studied by Pillai in 1933) and $\widetilde{P}(n):=n \prod_{p\vert n}(2-1/p)$ (studied by Toth in 2009). From their results, one can obtain asymptotic expansions for $\sum_{n\le x} P(n)/n$ and $\sum_{n\le x} \widetilde{P}(n)/n$. We consider two wide classes of functions ${\mathcal R}$ and ${\mathcal U}$ of arithmetical functions which include $P(n)/n$ and $\widetilde{P}(n)/n$ respectively. For any given $R\in {\mathcal R}$ and $U\in {\mathcal U}$, we obtain asymptotic expansions for $\sum_{n\le x} R(n), \sum_{n\le x} U(n), \sum_{p\le x} R(p-1)$ and $\sum_{p\le x} U(p-1)$.
Classification : 11A25, 11N37
Keywords: gcd-sum function, Dirichlet divisor problem, shifted primes
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     title = {Some remarks on a paper of {L.} {T\'oth}},
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De Koninck, Jean-Marie; Kátai, Imre. Some remarks on a paper of L. Tóth. Journal of integer sequences, Tome 13 (2010) no. 1. http://geodesic.mathdoc.fr/item/JIS_2010__13_1_a0/