Highly Composite Numbers by Srinivasa Ramanujan

  title={Highly Composite Numbers by Srinivasa Ramanujan},
  author={Jean-Louis Nicolas and Guy Robin},
  journal={The Ramanujan Journal},
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 But it was not the whole work on the subject, and in “The lost notebook and other unpublished papers”, one can find a manuscript, handwritten by Ramanujan, which is the continuation of the paper published by the London Mathematical Society.This paper is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… 

Highly Composite Numbers

In 1915, the London Mathematical Society published in its Proceedings a paper by Ramanujan entitled Highly Composite Numbers. A number N is said to be highly composite if for every integer M<N, it

Prime Numbers and Highly Composite Numbers

In 1915, Ramanujan wrote a long paper on “highly composite numbers.” This paper gives us a general method to analyse the growth of arithmetic functions. It is curious that this paper finds no

On SA, CA, and GA numbers

Gronwall’s function G is defined for n>1 by $G(n)=\frac{\sigma(n)}{n \log\log n}$ where σ(n) is the sum of the divisors of n. We call an integer N>1 a GA1 number if N is composite and G(N)≥G(N/p) for


Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivic, Schwarz, Wirsing, and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert's

Superabundant numbers, their subsequences and the Riemann hypothesis

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n) 5040 (\gamma is Euler's constant). It is a

Fast finite field arithmetic

The present work aims to study the implications of the new complexity bound, from a theoretical and practical point of view, and presents two new algorithms that should make the computation faster in practice (rather than asymptotically speaking).

An Elementary Problem Equivalent to the Riemann Hypothesis

The function a (n) = dInd is the sum-of-divisors function, so for example a (6) = 12. The number Hn is called the nth harmonic number by Knuth, Graham, and Patashnik [12, sect. 6.3], who detail

On the Number of Factorizations of Polynomials over Finite Fields

  • Rachel N. BermanR. Roth
  • Mathematics, Computer Science
    2020 IEEE International Symposium on Information Theory (ISIT)
  • 2020
For the two enumeration problems, bounds are obtained on the maximum number of factorizations, and a characterization is presented for polynomials attaining that maximum.

Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis

It is proved that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) ≥ max(G(N/p), G(aN)), for all prime factors p of N and each positive integer a.

Pseudorandom tableau sequences

It is shown that BPT sequences have excellent autocorrelation properties and the starting number of the tableau sequence may be arbitrary, which increases the complexity of the sequence.



On Highly Composite Numbers

for a certain c. In fact I shall prove that if n is highly composite, then the next highly composite number is less than n+n(log y&)-C ; and the result just stated follows immediately from this. At,

The lost notebook and other unpublished papers

Upper bounds for sums of powers of divisor functions

A look back at Ramanujan's Notebooks

  • B. Birch
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1975
Ramanujan's notebooks were the theme of a lecture (20) given by G. N. Watson to the London Mathematical Society in 1931. At that time, he and B. M. Wilson were collaborating on a critical edition;

Petites valeurs de la fonction d'Euler

Collected Papers

THIS volume is the first to be produced of the projected nine volumes of the collected papers of the late Prof. H. A. Lorentz. It contains a number of papersnineteen in all, mainly printed

Majorations Explicites Pour le Nombre de Diviseurs de N

Abstract Let It is proved that the function f reaches its maximum for n = 6 983 776 800, and that maxn≥2 f(n) = 1.5379. The proof deals with superior highly composite numbers introduced by Ramanujan.

Répartition Des Nombres Hautement Composés de Ramanujan

  • J. Nicolas
  • Mathematics
    Canadian Journal of Mathematics
  • 1971
On dit qu'un nombre entier A est hautement composé si tout nombre M plus petit que A a moins de diviseurs que A. Si l'on définit d(n) = nombre de diviseurs de n, on sait que, si la décomposition en

Répartition des nombres largement composés

© Séminaire Delange-Pisot-Poitou. Théorie des nombres (Secrétariat mathématique, Paris), 1977-1978, tous droits réservés. L’accès aux archives de la collection « Séminaire Delange-Pisot-Poitou.

Répartition des nombres superabondants

L’accès aux archives de la revue « Bulletin de la S. M. F. » ( http://smf. emath.fr/Publications/Bulletin/Presentation.html), implique l’accord avec les conditions générales d’utilisation (