Multiplicative partitions of numbers with a large squarefree divisor

Abstract

For each positive integer n, let f(n) denote the number of multiplicative partitions of n, meaning the number of ways of writing n as a product of integers larger than 1, where the order of the factors is not taken into account. It was shown by Oppenheim (J Lond Math Soc 1:205–211, 1926) that, as $$x\rightarrow \infty $$ , $$\begin{aligned} \max _{\begin{array}{c} n \le x \\ n\text { squarefree} \end{array}} f(n) = x/L(x)^{2+o(1)}, \end{aligned}$$ where $$L(x) = \exp \big (\log x\cdot \frac{\log \log \log {x}}{\log \log x}\big )$$ . Without the restriction to squarefree n, the maximum is the significantly larger quantity $$x/L(x)^{1+o(1)}$$ ; this was proved by Canfield et al. (J Number Theory 17:1–28, 1983). We prove the following theorem that interpolates between these two results: for each fixed $$\alpha \in [0,1]$$ , $$\begin{aligned} \max _{\begin{array}{c} n \le x\\ \mathrm {rad}(n) \ge n^{\alpha } \end{array}} f(n) = x/L(x)^{1+\alpha +o(1)}. \end{aligned}$$ We deduce, on the abc conjecture, a nontrivial upper bound on how often values of certain polynomials appear in the range of Euler’s $$\varphi $$ -function.