Abstract:

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