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The divisors of p-1

Published online by Cambridge University Press:  26 February 2010

R. R. Hall
R. R. Hall
Affiliation:
Department of Mathematics, University of York, Heslington, York.
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Extract

Let τ(n; k, h) denote the number of divisors of n which are congruent to h (mod k), and τ(n; k) the number of divisors of n prime to k, so that

Let

Erdős [1] proved that, if ε and η are fixed arbitrary positive numbers, then for almost all integers n ≤ x, we have

provided

Hall and Sudbery [2] showed that it is sufficient that

and apart from the ε, this upper bound for k is best possible, for it is clear that k must not exceed the normal order of τ(n). For n ≤ x, Hardy and Ramanujan [3] showed that this normal order is (log x)log 2 = 2 log log x.

Type
Research Article
Information
Mathematika , Volume 20 , Issue 1 , June 1973 , pp. 87 - 97
Copyright
Copyright © University College London 1973

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References

1. Erdos, P., “On the distribution of divisors of integers in the residue classes (mod d)”, Bull. Math. Soc. Grece, 6 (1965), 27–36.Google Scholar
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3. Hardy, G. H. and Ramanujan, S., “The normal number of prime factors of a number n”, Quarterly Journal, 48 (1917), 76–92.Google Scholar
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6. Ramanujan, S., “Some formulae in the analytic theory of numbers”, Messenger of Mathematics, 45 (1915), 81–84.Google Scholar
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8. Titchmarsh, E. C., The theory of the Riemann zeta-function (Oxford, 1951).Google Scholar