He discovered fewer new things in analysis, possibly because he lacked the formal education and did not find books to learn it from, but rediscovered many results, including the prime number theorem. In analysis, he worked on the elliptic functions and the analytic theory of numbers. In analytic number theory, he was as imaginative as usual, but much of what he imagined was wrong. Hardy blamed this on the inherent difficulty of analytic number theory, where imagination had led many great mathematicians astray. In analytic number theory, rigorous proof is more important than imagination, the opposite of Ramanujan's style. His "one great failure" is that he knew "nothing at all about the theory of analytic functions".
He discovered fewer new things in analysis, possibly because he lacked the formal education and did not find books to learn it from, but rediscovered many results, including the prime number theorem. In analysis, he worked on the elliptic functions and the analytic theory of numbers. In analytic number theory, he was as imaginative as usual, but much of what he imagined was wrong. Hardy blamed this on the inherent difficulty of analytic number theory, where imagination had led many great mathematicians astray. In analytic number theory, rigorous proof is more important than imagination, the opposite of Ramanujan's style. His ".one great failure" is that he knew "nothing at all about the theory of analytic functions"