New Algorithms For Computing The Riemann Zeta Function

The most efficient elgorithm known for computing the Riemann zeta function is based on the Riemann-Siegel formula, and requires approximately T1/2 operations when the argument is at height T. This algorithm has been used for some very extensive computations, which typically compute a very large number of values in a small neighborhood. In those situations, a much more efficient procedure has been developed. It relies on the Fast Fourier Transform and novel techniques for evaluation of rational functions to evaluate a set of neighboring values of the zeta function at average cost of O(tepsilon).