Ramanujan’s Theory of Prime Numbers

In his famous letters of 16 January 1913 and 29 February 1913 to G. H. Hardy, Ramanujan [23, pp. xxiii-xxx, 349–353] made several assertions about prime numbers, including formulas for π(x), the number of prime numbers less than or equal to x. Some of those formulas were analyzed by Hardy [3], [5, pp. 234–238] in 1937. A few years later, Hardy [7, Chapter II], in a very penetrating and lucid presentation, thoroughly discussed most of the results on primes found in these letters. In particular, Hardy related Ramanujan’s fascinating, but unsound, argument for deducing the prime number theorem. Generally, Ramanujan thought that his formulas for π(x) gave better approximations than they really did. As Hardy [7, p. 19] (Ramanujan [23, p. xxiv]) pointed out, some of Ramanujan’s faulty thinking arose from his assumption that all of the zeros of the Riemann zeta-function ζ(s) are real.