Multiplicatively monotonous arithmetic functions (Q2318635)

A function of \(f: \mathbb{N}\to\mathbb{R}\) is called multiplicatively growing, if \[ k\mid n\Rightarrow f(k)\leq f(n) \] (similarly multiplicatively decreasing and multiplicatively monotone). Classical examples of multiplicatively growing functions are the characteristic functions of sets of multiples, studied by \textit{H. Davenport} and \textit{P. Erdős} [Acta Arith. 2, 147--151 (1936; Zbl 0015.10001; JFM 62.1130.03)] and \textit{P. Erdős} [Bull. Am. Math. Soc. 54, 685--692 (1948; Zbl 0032.01301)]. The author generalizes a result of Erdős, that these functions have a logarithmic mean-value. He applies this to positive hermitian Toeplitz-multiplicative determinants, whose successive quotients also generate a multiplicative decreasing function.