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Highly Composite Numbers

  • Chapter
Ramanujan's Lost Notebook

Abstract

In 1915, the London Mathematical Society published in its Proceedings a paper by Ramanujan entitled Highly Composite Numbers. A number N is said to be highly composite if for every integer M<N, it happens that d(M)<d(N), where d(n) is the number of divisors of n. In the notes of Ramanujan’s Collected Papers, the editors relate, “The paper, long as it is, is not complete.” Fortunately, the large remaining portion of the paper was not discarded. It was first set into print by Jean-Louis Nicolas and Guy Robin in the first volume of the Ramanujan Journal, for which they provided useful comments. This chapter contains that formerly unpublished completion of Ramanujan’s paper as well as updated annotations.

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Notes

  1. 1.

    Since the first value of ρ(1−ρ) is about 200 we see that the geometric mean is a much closer approximation than either.

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Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, USA

    George E. Andrews

  2. Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, IL, USA

    Bruce C. Berndt

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  1. George E. Andrews
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  2. Bruce C. Berndt
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Correspondence to George E. Andrews .

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Andrews, G.E., Berndt, B.C. (2012). Highly Composite Numbers. In: Ramanujan's Lost Notebook. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3810-6_10

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