Abstract
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 But it was not the whole work on the subject, and in “The lost notebook and other unpublished papers”, one can find a manuscript, handwritten by Ramanujan, which is the continuation of the paper published by the London Mathematical Society.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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Alaoglu and P. Erdös, “On highly composite and similar numbers,” Trans. Amer. Math. Soc. 56(1944), 448-469.
B.J. Birch, “A look back at Ramanujan's notebook,” Math. Proc. Camb. Phil. Soc. 78(1975), 73-79.
J.L. Duras, J.L. Nicolas, and G. Robin, “Majoration des fonctions d k(n),” (to appear).
P. Erdös and J.L._ Nicolas, “Répartition des nombres superabondants,” Bull. Soc. Math. France 103(1975), 113-122.
G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford, 1971.
G.H. Hardy, Ramanujan, Cambridge Univ. Press, 1940 and Chelsea, 1978.
J.L. Nicolas, “Répartition des nombres hautement composés de Ramanujan,” Canadian J. Math. 23(1971), 115-130.
J.L. Nicolas, “Grandes Valeurs Des Fonctions Arithmétiques,” Séminaire D.P.P. Paris, (16e année, 1974/75), n o. G20, p. 5.
J.L. Nicolas, “Répartition des nombres largement composés,” Acta Arithmetica 34(1980), 379-390.
J.L. Nicolas and G. Robin, “Majorations explicites pour le nombre de diviseurs de N,” Canad. Math. Bull. 26(1983), 485-492.
J.L. Nicolas, “Petites valeurs de la fonction d'Euler,” Journal of Number Theory 17(1983), 375-388.
J.L. Nicolas, “On Highly Composite Numbers,” Ramanujan revisited, Academic Press, 1988, pp. 216-244.
J.L. Nicolas, “On Composite Numbers,” Number Theory, Madras 1987, Lecture Notes in Maths. 1395, edited by K. Alladi, Springer Verlag, 1989, pp. 18-20.
K.K. Norton, “Upper bounds for sums of powers of divisor functions,” J. Number Theory 40(1992), 60-85.
H. Rademacher, Topics in Analytic Number Theory, Springer Verlag, 1973.
S. Ramanujan, “Highly Composite Numbers,” Proc. London Math. Soc. Serie 2 14(1915), 347-400.
S. Ramanujan, Collected Papers, Cambridge University Press, 1927, and Chelsea 1962.
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House and Springer Verlag, New Delhi, 1988.
R.A. Rankin, “Ramanujan's manuscripts and notebooks II,” Bull. London Math. Soc. 21(1989), 351-365.
G. Robin, Sur l'ordre maximum de la fonction somme des diviseurs, Séminaire Delange-Pisot-Poitou. Paris, 1981-1982; Progress in MathematicsBirkhäuser, 38(1983), 223-244.
G. Robin, Grandes valeurs de fonctions arithmétiques et problèmes d'optimisation en nombres entiers, Thèse d'Etat, Université de Limoges, France, 1983.
G. Robin, “Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann,” J. Math. Pures et Appl. 63(1984), 187-213.
G. Robin, “Sur la différence Li(?(x)) -??x?,” Ann. Fac. Sc. Toulouse 6(1984), 257-268.
G. Robin, “Grandes valeurs de la fonction somme des diviseurs dans les progressions arithmétiques,” J. Math. Pures et Appl. 66(1987), 337-349.
G. Robin, Sur des Travaux non publiés de S. Ramanujan sur les Nombres Hautement Composés, Publications du département de Mathématiques de l'Université de Limoges, France, 1991, pp. 1-60.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nicolas, JL., Robin, G. Highly Composite Numbers by Srinivasa Ramanujan. The Ramanujan Journal 1, 119–153 (1997). https://doi.org/10.1023/A:1009764017495
Issue Date:
DOI: https://doi.org/10.1023/A:1009764017495