Multiplicative partitions of numbers with a large squarefree divisor
For a positive integer \(n\), write \(f(n)\) for the number of multiplicative partitions of \(n\), i.e., the number of decompositions of \(n\) as a product of integers larger than \(1\), where two decompositions are considered to be equal if they differ only by the order of the factors. Writing \[ L(x) = \exp\left(\log x \frac{\log_3 x}{\log_2 x}\right), \] \textit{E. R. Canfield} et al. [J. Number Theory 17, 1--28 (1983; Zbl 0513.10043)] proved that \[ \max_{n \leq x} f(n) = x/L(x)^{1+o(1)}. \] Previous work of \textit{A. Oppenheim} [J. Lond. Math. Soc. 1, 205--211 (1926; JFM 52.0163.02)] implied (after a correction) that \[ \max_{\substack{n \leq x \\ n \text{ square-free}}} f(n) = x/L(x)^{2+o(1)}. \] The paper under review interpolates between these two results by showing that \[ \max_{\substack{n \leq x \\ \operatorname{rad}(n) \geq n^\alpha}} f(n) = x/L(x)^{1+\alpha+o(1)}, \] where \(\operatorname{rad}(n)\) is the radical of \(n\), i.e. the largest square-free divisor of \(n\). The author's orginal motivation to study this is an application concerning value distribution of the Euler \(\varphi\)-function. Writing \(F(n) = \# \varphi^{-1}(n)\) for the number of pre-images of \(n\), the author considers \[ \max_{0 < |P(n)| \leq x} F(|P(n)|) \] with \(P(n)\) a non-zero polynomial with at least two distinct complex roots. Assuming the abc conjecture, the author shows (in a precise sense) that this maximum is of smaller order of magnitude than a known (conditional) asymptotic estimate for \[ \max_{n \leq x} F(n), \] i.e. he shows that \(P(n)\) cannot be as popular value of \(\varphi(n)\) as possible.
scientific article; zbMATH DE number 7345289
scientific article