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2012 | OriginalPaper | Chapter

Highly Composite Numbers

Authors : George E. Andrews, Bruce C. Berndt

Published in: Ramanujan's Lost Notebook

Publisher: Springer New York

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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 This chapter contains that formerly unpublished completion of Ramanujan’s paper as well as updated annotations.

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previous chapter Circular Summation
Footnotes
1
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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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Metadata
Title
Highly Composite Numbers
Authors
George E. Andrews
Bruce C. Berndt
Copyright Year
2012
Publisher
Springer New York
Book
Ramanujan's Lost Notebook

Print ISBN: 978-1-4614-3809-0

Electronic ISBN: 978-1-4614-3810-6

Copyright Year: 2012

DOI
https://doi.org/10.1007/978-1-4614-3810-6_10

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