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VARIATIONS ON A THEOREM OF DAVENPORT CONCERNING ABUNDANT NUMBERS

Published online by Cambridge University Press:  12 September 2013

EMILY JENNINGS
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA email [email protected]@math.uga.edu
PAUL POLLACK*
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA email [email protected]@math.uga.edu
LOLA THOMPSON
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA email [email protected]@math.uga.edu
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Abstract

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Let $\sigma (n)= {\mathop{\sum }\nolimits}_{d\mid n} d$ be the usual sum-of-divisors function. In 1933, Davenport showed that $n/ \sigma (n)$ possesses a continuous distribution function. In other words, the limit $D(u): = \lim _{x\rightarrow \infty }(1/ x){\mathop{\sum }\nolimits}_{n\leq x, n/ \sigma (n)\leq u} 1$ exists for all $u\in [0, 1] $ and varies continuously with $u$. We study the behaviour of the sums ${\mathop{\sum }\nolimits}_{n\leq x, n/ \sigma (n)\leq u} f(n)$ for certain complex-valued multiplicative functions $f$. Our results cover many of the more frequently encountered functions, including $\varphi (n)$, $\tau (n)$ and $\mu (n)$. They also apply to the representation function for sums of two squares, yielding the following analogue of Davenport’s result: for all $u\in [0, 1] $, the limit

$$\begin{eqnarray*}\tilde {D} (u): = \lim _{R\rightarrow \infty }\frac{1}{\pi R} \# \biggl\{ (x, y)\in { \mathbb{Z} }^{2} : 0\lt {x}^{2} + {y}^{2} \leq R\text{ and } \frac{{x}^{2} + {y}^{2} }{\sigma ({x}^{2} + {y}^{2} )} \leq u\biggr\}\end{eqnarray*}$$
exists, and $\tilde {D} (u)$ is both continuous and strictly increasing on $[0, 1] $.

Type
Research Article
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Billingsley, P., Probability and Measure, 3rd edn. Wiley Series in Probability and Mathematical Statistics (John Wiley & Sons, New York, 1995).Google Scholar
Davenport, H., ‘Über numeri abundantes’, S.-Ber. Preuß. Akad. Wiss., math.-nat. Kl. (1933), 830837.Google Scholar
Delange, H., ‘Sur les fonctions arithmétiques multiplicatives’, Ann. Sci. Éc. Norm. Supér. (3) 78 (1961), 273304.CrossRefGoogle Scholar
Elliott, P. D. T. A., Probabilistic number theory I: mean-value theorems, Grundlehren der Mathematischen Wissenschaften, 239 (Springer, New York-Berlin, 1979).Google Scholar
Erdős, P. and Wintner, A., ‘Additive arithmetical functions and statistical independence’, Amer. J. Math. 61 (1939), 713721.Google Scholar
Halász, G., ‘Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen’, Acta Math. Acad. Sci. Hungar. 19 (1968), 365403.Google Scholar
Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 6th edn. (Oxford University Press, Oxford, 2008).CrossRefGoogle Scholar
Kobayashi, M., On the Density of Abundant Numbers. PhD Thesis, Dartmouth College. 2010.Google Scholar
Moree, P., ‘Counting numbers in multiplicative sets: Landau versus Ramanujan’, Math. Newsl. 21 (2011), 7381.Google Scholar
Pillai, S. S., ‘Generalisation of a theorem of Mangoldt’, Proc. Indian Acad. Sci., Sect. A. 11 (1940), 1320.CrossRefGoogle Scholar
Schwarz, W. and Spilker, J., Arithmetical Functions, London Mathematical Society Lecture Note Series, 184 (Cambridge University Press, Cambridge, 1994).CrossRefGoogle Scholar
Wirsing, E., ‘Das asymptotische Verhalten von Summen über multiplikative Funktionen. II’, Acta Math. Acad. Sci. Hungar. 18 (1967), 411467.Google Scholar