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 when is negative and fractional to conform to the law which holds when is a positive integer, similarly the whole of my investigations proceed on giving a meaning to Eulerian Second Integral for all values of . My friends who have gone through the regular course of University education tell me that is true only when is positive. They say that this integral relation is not true when is negative. Supposing this is true only for positive values of and also supposing the definition 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 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