Mathematics · 1913

Letter to G. H. Hardy, 16 January 1913

Srinivasa Ramanujan

Excerpt · The covering letter of Ramanujan’s first letter to Hardy

As printed in the Collected Papers of Srinivasa Ramanujan, Cambridge University Press, 1927, p. xxiii

Source: archive.org
Retrieved: 27 September 2026
License: Public domain
Rights holder: Srinivasa Ramanujan, 1913
Language: English

About this edition and its rights

Edition

The letter as printed in Collected Papers of Srinivasa Ramanujan, edited by G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson (Cambridge University Press, 1927), p. xxiii, in Hardy’s notice of Ramanujan, first printed in the Proceedings of the London Mathematical Society (2), xix (1921). Kept: the letter itself, complete as printed there, from “Madras, 16th January 1913” to the postscript; left out: Hardy’s text around it and the extracts from “the papers enclosed” and from later letters that he prints after it. The text comes from the OCR of the Internet Archive item pli.kerala.rare.28155 (Kerala State Library copy), corrected on the scan of that page and compared with the OCR of a second scan of the same edition on the Internet Archive; the line with the integral, which both OCRs garble, and the formulas were set from the scan. Words hyphenated at line ends are rejoined; the formulas and the italic n are set with KaTeX and italics; the place and date, the salutation and the signature, set in small capitals as printed, keep their lines.

Rights

Srinivasa Ramanujan died on 26 April 1920 (Wikidata Q83163); the letter was first printed in 1921 and in this edition in 1927. It is in the public domain in France, where the author died before 1 January 1956, and in the United States, where it was published before 1 January 1931. Hardy (died 1 December 1947, Wikidata Q184337), who printed it, would also be in the public domain in France, but none of his text is reproduced.

Hardy writes, just before the letter, that Ramanujan’s letters were probably not entirely his own and that friends gave him some assistance with the English; those friends are not named.

Madras, 16th January 1913.

Dear Sir,

I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £20 per annum. I am now about 23 years of age. I have had no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling”.

Just as in elementary mathematics you give a meaning to $a^{n}$ when n is negative and fractional to conform to the law which holds when n is a positive integer, similarly the whole of my investigations proceed on giving a meaning to Eulerian Second Integral for all values of n. My friends who have gone through the regular course of University education tell me that $\int_{0}^{\infty} x^{n - 1} e^{- x} d x = Γ (n)$ is true only when n is positive. They say that this integral relation is not true when n is negative. Supposing this is true only for positive values of n and also supposing the definition $n Γ (n) = Γ (n + 1)$ to be universally true, I have given meanings to these integrals and under the conditions I state the integral is true for all values of n negative and fractional. My whole investigations are based upon this and I have been developing this to a remarkable extent so much so that the local mathematicians are not able to understand me in my higher flights.

Very recently I came across a tract published by you styled Orders of Infinity in page 36 of which I find a statement that no definite expression has been as yet found for the number of prime numbers less than any given number. I have found an expression which very nearly approximates to the real result, the error being negligible. I would request you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you.

I remain, Dear Sir, Yours truly,
S. Ramanujan.

P.S. My address is S. Ramanujan, Clerk Accounts Department, Port Trust, Madras, India.