On a number-theoretic function (Q2724028)

The authors investigate, for fixed \(\alpha \in \mathbb R\), the multiplicative arithmetical function NEWLINE\[NEWLINE \psi_\alpha(n) := \sum_{i=1}^n \left({n\over(i,n)}\right)^\alpha = \sum_{d|n}d^\alpha\varphi(d), NEWLINE\]NEWLINE 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).NEWLINENEWLINENEWLINEReviewer'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 \(\sum_{i=1}^\alpha ({n\over(i,n)})^n\).