The Ramanujan Journal

, Volume 1, Issue 2, pp 119–153

Highly Composite Numbers by Srinivasa Ramanujan

  • Jean-Louis Nicolas
  • Guy Robin

DOI: 10.1023/A:1009764017495

Cite this article as:
Nicolas, JL. & Robin, G. The Ramanujan Journal (1997) 1: 119. doi:10.1023/A:1009764017495


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 But it was not the whole work on the subject, and in “The lost notebook and other unpublished papers”, one can find a manuscript, handwritten by Ramanujan, which is the continuation of the paper published by the London Mathematical Society.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

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Jean-Louis Nicolas
    • 1
  • Guy Robin
    • 2
  1. 1.Institut Girard Desargues, UPRES-A-5028, MathématiquesUniversité Claude Bernard (LYON1)Villeurbanne cédexFrance
  2. 2.UPRES-A-6090, Théorie des nombres, calcul formel et optimisationUniversité de LimogesLimoges cédexFrance