#### Title

Variations on a theorem of Davenport concerning abundant numbers

#### Abstract

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)≤uf(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 D~(u):=limR→∞1πR#{(x,y)∈Z2:0

#### Repository Citation

Jennings, E., P. Pollack, and L. Thompson. 2014. "Variations on a theorem of Davenport concerning abundant numbers." Bulletin of the Australian Mathematical Society 89(3): 437-450.

#### Publisher

Cambridge University Press

#### Publication Date

1-1-2014

#### Publication Title

Bulletin of the Australian Mathematical Society

#### Department

Mathematics

#### Document Type

Article

#### DOI

10.1017/S0004972713000695

#### Keywords

Abundant number, Distribution function, Mean values of multiplicative functions, Sum-of-divisors function

#### Language

English

#### Format

text