John Littlewood on Srinivasa Ramanujan

“I read in the proof-sheets of Hardy on Ramanujan: 'As someone said, each of the positive integers was one of his personal friends.' My reaction was, 'I wonder who said that; I wish I had.' In the next proof- sheets I read (what now stands), 'It was Littlewood who said... '”

“Ramanujan's great gift is a 'formal' one; he dealt in 'formulae'. To be quite clear what is meant, I give two examples (the second is at random, the first is one of supreme beauty):p(4)+p(9) x+p(14) x^2+ldots=5 frac{left{left(1-x^5right)left(1-x^{10}right)left(1-x^{15}right) ldotsright}^5}{left{(1-x)left(1-x^2right)left(1-x^3right) ldotsright}^6} where p(n) is the number of partitions of n; ... But the great day of formulae seems to be over. No one, if we are again to take the highest standpoint, seems able to discover a radically new type, though Ramanujan comes near it in his work on partition series; it is futile to multiply examples in the spheres of Cauchy's theorem and elliptic function theory, and some general theory dominates, if in a less degree, every other field. A hundred years or so ago his powers would have had ample scope... The beauty and singularity of his results is entirely uncanny... the reader at any rate experiences perpetual shocks of delighted surprise. And if he will sit down to an unproved result taken at random, he will find, if he can prove it at all, that there is at lowest some 'point', some odd or unexpected twist. ... His intuition worked in analogies, sometimes remote, and to an astonishing extent by empirical induction from particular numerical cases... his most important weapon seems to have been a highly elaborate technique of transformation by means of divergent series and integrals. (Though methods of this kind are of course known, it seems certain that his discovery was quite independent.) He had no strict logical justification for his operations. He was not interested in rigour, which for that matter is not of first-rate importance in analysis beyond the undergraduate stage, and can be supplied, given a real idea, by any competent professional.”