Variations on a theorem of Davenport concerning abundant numbers

Abstract: Let \sigma(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/\sigma(n) possesses a continuous distribution function. In other words, the limit D(u):= \lim_{x\to\infty} \frac{1}{x}\sum_{n \leq x,~n/\sigma(n) \leq u} 1 exists for all u \in [0,1] and varies continuously with u. We study the behavior of the sums \sum_{n \leq x,~n/\sigma(n) \leq u} f(n) for certain complex-valued multiplicative functions f. Our results cover many of the more frequently encountered functions, including \varphi(n), \tau(n), and \mu(n). They also apply to the representation function for sums of two squares, yielding the following analogue of Davenport's result: For all u \in [0,1], the limit [ \tilde{D}(u):= \lim_{R\to\infty} \frac{1}{\pi R}#{(x,y) \in \Z^2: 0<x^2+y^2 \leq R \text{ and } \frac{x^2+y^2}{\sigma(x^2+y^2)} \leq u} ] exists, and \tilde{D}(u) is both continuous and strictly increasing on [0,1].