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The Divisibility of Divisor Functions

Published online by Cambridge University Press:  18 May 2009

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 nx 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
Copyright
Copyright © Glasgow Mathematical Journal Trust 1961

References

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2.Hardy, G. H., Ramanujan (Cambridge, 1940).Google Scholar
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5.Walfisz, A., Über einige Ramanujansche Sätze, Trav. Inst. Math. Tbilissi 5 (1938), 145152.Google Scholar
6.Watson, G. N., Uber Ramanujansche Kongruenzeigenschaften der Zerfāllungsanzahlen (I), Math. Z. 39 (1935), 712731.CrossRefGoogle Scholar