Variations on a Theorem of Davenport concerning Abundant Numbers

Let σ(n) = ∑ d|nd be the usual sum-of-divisors function. In 1933, Davenport showed that n/σ(n) possesses a continuous distribution function. In other words, the limit D(u) := limx→∞(1/x) ∑ n≤x,n/σ(n)≤u1 exists for all u ∈ [0, 1] and varies continuously with u. We study the behaviour of the sums ∑n≤x,n/σ(n)≤u f (n) for certain complex-valued multiplicative functions f . Our results cover many of the more frequently encountered functions, including φ(n), τ(n) and μ(n). They also apply to the representation function for sums of two squares, yielding the following analogue of Davenport’s result: for all u ∈ [0, 1], the limit D̃(u) := lim R→∞ 1 πR # { (x, y) ∈ Z : 0 < x + y ≤ R and x 2 + y2 σ(x2 + y2) ≤ u } exists, and D̃(u) is both continuous and strictly increasing on [0, 1]. 2010 Mathematics subject classification: primary 11N60; secondary 11A25, 11N37.