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THE ERROR TERM IN THE COUNT OF ABUNDANT NUMBERS

Part of: Multiplicative number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms Elementary number theory

Published online by Cambridge University Press:  02 January 2014

Mitsuo Kobayashi  and
Paul Pollack
Mitsuo Kobayashi
Affiliation:
Department of Mathematics and Statistics, Cal Poly Pomona, Pomona, CA 91768, U.S.A. email [email protected]
Paul Pollack
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602,U.S.A. email [email protected]
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Abstract

A natural number $n$ is called abundant if the sum of the proper divisors of $n$ exceeds $n$. For example, $12$ is abundant, since $1+ 2+ 3+ 4+ 6= 16$. In 1929, Bessel-Hagen asked whether or not the set of abundant numbers possesses an asymptotic density. In other words, if $A(x)$ denotes the count of abundant numbers belonging to the interval $[1, x] $, does $A(x)/ x$ tend to a limit? Four years later, Davenport answered Bessel-Hagen’s question in the affirmative. Calling this density $\Delta $, it is now known that $0. 24761\lt \Delta \lt 0. 24766$, so that just under one in four numbers are abundant. We show that $A(x)- \Delta x\lt x/ \mathrm{exp} (\mathop{(\log x)}\nolimits ^{1/ 3} )$ for all large $x$. We also study the behavior of the corresponding error term for the count of so-called $\alpha $-abundant numbers.

MSC classification

Secondary: 11A25: Arithmetic functions; related numbers; inversion formulas 11K65: Arithmetic functions 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Mathematika , Volume 60 , Issue 1 , January 2014 , pp. 43 - 65
Copyright
Copyright © University College London 2014 

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