Supercomputers and the Riemann Zeta Function

The riemann Hypothesis, which specifies the location of zeros of the Riemann zeta function, and thus describes the behavior of primes, is one of the most famous unsolved problems in mathematics, and extensive efforts have been made over more than a century to check it numerically for large sets of cases. Recently a new algorithm, invented by the speaker and A. Schonhage, has been implemented, and used to compute over 175 million zeros near zero number 10 sip 20. The new algorithm turned out to be over 5 orders of magnitude faster than older methods. The crucial ingredients in it are a rational function evaluation method similar to the Greengard-Rokhlin gravitational potential evaluation algorithm, the FFT, and band-limited function interpolation. While the only present implementation is on a Cray, the algorithm can easily be parallelized.