Sándor, J.; Kramer, A.-V. On a number-theoretic function. (Über eine zahlentheoretische Funktion.) (German) Zbl 0972.11005 Math. Morav. 3, 53-62 (1999). The authors investigate, for fixed \(\alpha \in \mathbb R\), the multiplicative arithmetical function \[ \psi_\alpha(n) := \sum_{i=1}^n \left({n\over(i,n)}\right)^\alpha = \sum_{d|n}d^\alpha\varphi(d), \] where \(\varphi(n)\) is Euler’s function, and give some motivation from group theory for introducing this function. Several elementary properties and inequalities concerning \(\psi_\alpha(n)\) are given, and a weak asymptotic formula for \(\sum_{n\leq x}\psi_\alpha(n)\) (which can be easily sharpened).Reviewer’s remark: There are no theorems in the paper, but there are several misprints, including the definition of \(\psi_\alpha(n)\) in Section 1 on p. 53, which is incorrectly written as \(\sum_{i=1}^\alpha ({n\over(i,n)})^n\). Reviewer: Aleksandar Ivić (Beograd) Cited in 1 Document MSC: 11A25 Arithmetic functions; related numbers; inversion formulas 11N37 Asymptotic results on arithmetic functions Keywords:arithmetic functions; multiplicative function; asymptotic formulas PDFBibTeX XMLCite \textit{J. Sándor} and \textit{A. V. Kramer}, Math. Morav. 3, 53--62 (1999; Zbl 0972.11005) Online Encyclopedia of Integer Sequences: Partial sums of Sum_{k=1..n} n/gcd(n,k), or partial sums of Sum_{d|n} d*phi(d) (see A057660).