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On A theorem of Niven

Published online by Cambridge University Press:  20 November 2018

Robert E. Dressler
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In [3], Niven proved that for any positive integer k, the density of the set of positive integers n for which (n, (φ(n))≤k is zero (where φ is the Euler to tient function). In this paper, we prove a related result—namely if k and j are any positive integers, then the density of the set of positive integers n for which (n,σj(n))≤k is zero (where σj(n) is the sum of the jth powers of the positive divisors of n). We will borrow from Niven’s technique, but we must make some crucial modifications.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 1 , 01 March 1974 , pp. 109 - 110
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Hardy, G. H. and Wright, E. M. An Introduction to the theory of numbers. Oxford University Press, Oxford, Fourth Edition (1960).Google Scholar
2. Veque, W. J. Le, Topics in number theory, Vol. I, Addison-Wesley Publishing Co. (1958).Google Scholar
3. Niven, L., The asymptotic density of sequences, Bull. A.M.S., 57 (1951), pp. 420-434.Google Scholar