Abstract

Consider the functions P (n) := ∑_{k=1}^{n} gcd(k, n) (studied by Pillai in 1933) and P̃(n) := n ∏_{p|n}(2−1/p) (studied by Toth in 2009). From their results, one can obtain asymptotic expansions for ∑_{n≤x} P (n)/n and ∑_{n≤x} P̃(n)/n. We consider two wide classes of functions R and U of arithmetical functions which include P (n)/n and P̃(n)/n respectively. For any given R ∈ R and U ∈ U, we obtain asymptotic expansions for ∑_{n≤x} R(n), ∑_{n≤x} U(n), ∑_{p≤x} R(p − 1) and ∑_{p≤x} U(p − 1).