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\begin{abstract}
Consider the functions $P(n):=\sum_{k=1}^n \gcd(k,n)$ (studied by
Pillai in 1933) and $\widetilde{P}(n):=n \prod_{p|n}(2-1/p)$
(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 $\sum_{p\le x} U(p-1)$.
\end{abstract}

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