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

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

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