Variations on a theorem of Davenport concerning abundant numbers

TL;DR

This paper extends Davenport's theorem on the distribution of the ratio n/σ(n) for abundant numbers, analyzing sums involving various multiplicative functions and applying results to sums of two squares, showing the limits are continuous and strictly increasing.

Contribution

It generalizes Davenport's distribution result to sums involving multiple multiplicative functions and the representation of numbers as sums of two squares.

Findings

The distribution functions for various multiplicative functions are continuous.

The limit distribution for sums of two squares exists and is strictly increasing.

Results apply to functions like φ(n), τ(n), and μ(n).

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].