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The Divisibility of Divisor Functions

Published online by Cambridge University Press:  18 May 2009

R. A. Rankin
R. A. Rankin
Affiliation:
The University Glasgow
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For any positive integers n and v let

where d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,

as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 5 , Issue 1 , January 1961 , pp. 35 - 40
Copyright
Copyright © Glasgow Mathematical Journal Trust 1961

References

1.Chowla, S., On a theorem of Walfisz, J. London Math. Soc. 22 (1947), 136–140.CrossRefGoogle Scholar
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3.Landau, E., Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Arch. Math. Phys. (3) 13 (1908), 305–312.Google Scholar
4.Landau, E., Handbuch der Lehre von der Verteilung der Primzahlen, Band 1 (Leipzig, 1909).Google Scholar
5.Walfisz, A., Über einige Ramanujansche Sätze, Trav. Inst. Math. Tbilissi 5 (1938), 145–152.Google Scholar
6.Watson, G. N., Uber Ramanujansche Kongruenzeigenschaften der Zerfāllungsanzahlen (I), Math. Z. 39 (1935), 712–731.CrossRefGoogle Scholar