The Ramanujan Journal

, Volume 1, Issue 2, pp 119-153

First online:

Highly Composite Numbers by Srinivasa Ramanujan

  • Jean-Louis NicolasAffiliated withInstitut Girard Desargues, UPRES-A-5028, Mathématiques, Université Claude Bernard (LYON1)
  • , Guy RobinAffiliated withUPRES-A-6090, Théorie des nombres, calcul formel et optimisation, Université de Limoges

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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 But it was not the whole work on the subject, and in “The lost notebook and other unpublished papers”, one can find a manuscript, handwritten by Ramanujan, which is the continuation of the paper published by the London Mathematical Society.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 number arithmetical function maximal order divisors