# Highly Composite Numbers by Srinivasa Ramanujan

@article{Nicolas1997HighlyCN,
title={Highly Composite Numbers by Srinivasa Ramanujan},
author={Jean-Louis Nicolas and Guy Robin},
journal={The Ramanujan Journal},
year={1997},
volume={1},
pages={119-153}
}
• Published 1 June 1997
• Mathematics
• 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 But it was not the whole work on the subject, and in “The lost notebook and other unpublished papers”, one can find a manuscript, handwritten by Ramanujan, which is the continuation of the paper published by the London Mathematical Society.This paper is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…
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## References

SHOWING 1-10 OF 19 REFERENCES

### 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,

### 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;

### 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

• Mathematics
• 1983
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.

• J. Nicolas
• Mathematics