VARIATIONS ON A THEOREM OF DAVENPORT CONCERNING ABUNDANT NUMBERS

EMILY JENNINGS, PAUL POLLACK, LOLA THOMPSON
2013 Bulletin of the Australian Mathematical Society  
Let σ(n) = ∑_d | nd be the usual sum-of-divisors function. In 1933, Davenport showed that that n/σ(n) possesses a continuous distribution function. In other words, the limit D(u):= _x→∞1/x∑_n ≤ x, n/σ(n) ≤ u 1 exists for all u ∈ [0,1] and varies continuously with u. We study the behavior 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
more » ... . 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):= _R→∞1/π R#{(x,y) ∈^2: 0<x^2+y^2 ≤ R and x^2+y^2/σ(x^2+y^2)≤ u} exists, and D̃(u) is both continuous and strictly increasing on [0,1].
doi:10.1017/s0004972713000695 fatcat:7ahrup3ph5ff7gtzly5suun3nm