Highly composite numbers. Annotated by Jean-Louis Nicolas and Guy Robin.

*(English)*Zbl 0917.11043The purpose and subject matter of this paper is well expressed in the Abstract: “In 1915, the London Mathematical Society published in its Proceedings a paper of Ramanujan entitled ‘Highly composite numbers’ [Proc. Lond. Math. Soc. (2) 14, 347–409 (1915; JFM 45.0286.02)]. But it was not the whole work on the subject, and in ‘The lost notebook and other unpublished papers’, one can find a manuscript, handwritten by Ramanujan, which is the continuation of the paper published by the London Mathematical Society.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 the works published after 1915 on the subject.

A number \(N\) is said to be 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 \(Q_{2k}(N)\) for \(1\leq k\leq 4\) where \(Q_{2k}(N)\) is the number of representations of \(N\) as a sum of \(2k\) squares and \(\sigma_{-s}(N)\) where \(\sigma_{-s}(N)\) is the sum of the \((-s)\)th powers of the divisors of \(N\). Moreover, the maximal orders of these functions are given.”

The foreword contains some background and the highlights of Ramanujan’s manuscript, which is reproduced in printed form in section 2. A table calculated by Ramanujan is displayed towards the end of the paper and alongside, for comparison purposes, is a table of largely composite numbers \(n\) (that satisfy \(d(m)\leq d(n)\) whenever \(m\leq n\)). The paper ends with notes detailing two omissions inserted and some minor corrections made and commenting on the main text and subsequent related work. Some of the results in section 2 have not appeared in print before. This paper underlines Ramanujan’s insight and manipulative ability, and leaves us some 85 years later still in awe of his skills.

A number \(N\) is said to be 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 \(Q_{2k}(N)\) for \(1\leq k\leq 4\) where \(Q_{2k}(N)\) is the number of representations of \(N\) as a sum of \(2k\) squares and \(\sigma_{-s}(N)\) where \(\sigma_{-s}(N)\) is the sum of the \((-s)\)th powers of the divisors of \(N\). Moreover, the maximal orders of these functions are given.”

The foreword contains some background and the highlights of Ramanujan’s manuscript, which is reproduced in printed form in section 2. A table calculated by Ramanujan is displayed towards the end of the paper and alongside, for comparison purposes, is a table of largely composite numbers \(n\) (that satisfy \(d(m)\leq d(n)\) whenever \(m\leq n\)). The paper ends with notes detailing two omissions inserted and some minor corrections made and commenting on the main text and subsequent related work. Some of the results in section 2 have not appeared in print before. This paper underlines Ramanujan’s insight and manipulative ability, and leaves us some 85 years later still in awe of his skills.

Reviewer: E.J.Scourfield (Egham)