Variations on a theorem of Davenport concerning abundant numbers

Title:
Variations on a theorem of Davenport concerning abundant numbers
Authors:
Jennings, Emily; Pollack, Paul; Thompson, Lola
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<x2+y2≤R and x2+y2σ(x2+y2)≤u} exists, and D~(u) is both continuous and strictly increasing on [0,1].
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
DATE ISSUED:
2014
Department:
Mathematics
Type:
Article
PUBLISHED VERSION:
10.1017/S0004972713000695
PERMANENT LINK:
http://hdl.handle.net/11282/566759

Full metadata record

DC FieldValue Language
dc.contributor.authorJennings, Emilyen
dc.contributor.authorPollack, Paulen
dc.contributor.authorThompson, Lolaen
dc.date.accessioned2015-08-13T10:33:24Zen
dc.date.available2015-08-13T10:33:24Zen
dc.date.issued2014en
dc.identifier.citationJennings, 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.en
dc.identifier.issn0004-9727en
dc.identifier.urihttp://hdl.handle.net/11282/566759en
dc.description.abstractLet 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<x2+y2≤R and x2+y2σ(x2+y2)≤u} exists, and D~(u) is both continuous and strictly increasing on [0,1].en
dc.language.isoen_USen
dc.publisherCambridge University Pressen
dc.identifier.doi10.1017/S0004972713000695en
dc.subject.departmentMathematicsen
dc.titleVariations on a theorem of Davenport concerning abundant numbersen
dc.typeArticleen
dc.identifier.journalBulletin of the Australian Mathematical Societyen
dc.subject.keywordAbundant numberen
dc.subject.keywordDistribution functionen
dc.subject.keywordMean values of multiplicative functionsen
dc.subject.keywordSum-of-divisors functionen
dc.identifier.volume89en
dc.identifier.issue3en
dc.identifier.startpage437en
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