Highly Composite Numbers by Srinivasa Ramanujan
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In 1915, the London Mathematical Society published in its Proceedings a paper of Ramanujan entitled “Highly Composite Numbers”. But it was not the whole work on the subject, and in “The lost notebook and other unpublished papers”, one can find a manuscript, handwritten by Ramanujan, which is the continuation of the paper published by the London Mathematical Society.
This paper is the typed version of the above mentioned manuscript with some notes, mainly explaining the link between the work of Ramanujan and works published after 1915 on the subject.
A number N is said highly composite if M < N implies d(M) < d(N), where d(N) is the number of divisors of N. In this paper, Ramanujan extends the notion of highly composite number to other arithmetic functions, mainly to Q2k (N) for 1 ≤ k ≤ 4 where Q2k (N) is the number of representations of N as a sum of 2k squares and σ-s(N) where σ-s(N) is the sum of the (-s)th powers of the divisors of N. Moreover, the maximal orders of these functions are given.
- Highly Composite Numbers by Srinivasa Ramanujan
The Ramanujan Journal
Volume 1, Issue 2 , pp 119-153
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- highly composite number
- arithmetical function
- maximal order
- Author Affiliations
- 1. Institut Girard Desargues, UPRES-A-5028, Mathématiques, Université Claude Bernard (LYON1), Bâtiment 101, F-69622, Villeurbanne cédex, France
- 2. UPRES-A-6090, Théorie des nombres, calcul formel et optimisation, Université de Limoges, F-87060, Limoges cédex, France