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HARDY TO RAMANUJAN
(EXCERPTS)
MAKE THE ACQUAINTANCE OF NEVILLE

 
Dear Mr. Ramanujan,
                                    
              . . . . your outline of the proof is too incomplete for me to pronounce a confident opinion, but it has all the appearance of being right, and it looks to me a very remarkable piece of work.

              
              
If you will send me your proof written out completely (so that it is easy to follow), I will (assuming that I agree with it of which I have very little doubt) try to get it published for you in England. Write it in the form of a paper. . . . giving a full proof of the principal and most remarkable theorem, . . . . .

               All this is correct. You infer (correctly) ... . You give no proof: but the result is true.

            
                You then infer that. . .. There is no theorem I know of which warrants such a conclusion: and I do not believe it to be a correct inference (though I cannot offhand construct an example to the contrary).


               Perhaps you have proved this: anyhow I am prepared to believe it.

               So I can make nothing of this step . . . Mr. Littlewood and I have proved this. ... But even here the proof is exceedingly difficult. I can see how your result is suggested: but to get a rigid proof is quite a different matter.

 

               You will see that, with all these gaps in the proof, it is no wonder that the result is wrong.

 


               The truth is that the theory of primes is full of pitfalls, to surmount which requires the fullest of trainings in modern rigorous methods. This you are naturally without. I hope you will not be discouraged by my criticisms. I think your argument a very remarkable and ingenious one. To have proved what you claimed to have proved would have been about the most remark able mathematical feat in the whole history of mathematics.

  

               As regards your work on continued fractions and elliptic functions - here the difficulties to be surmounted are of an entirely different kind, and I have no reason at all to suppose that your results are not perfectly correct. I hope you will adopt the suggestions I made at the beginning.

  

               Try to make the acquaintance of Mr. E.H. Neville, who is now in Madras lecturing. He comes from my college and you might find his advice as to reading and study invaluable.

  

               Well, you will see from the length of this letter that answering yours is not entirely trifling business and that I have some excuse if I have delayed.

 

Believe me
 

Yours very sincerely
G. H. Hardy.