Fast methods to compute the Riemann zeta function

By G. Hiary. 2007

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The Riemann zeta function on the critical line can be computed using a straightforward application of the Riemann-Siegel formula, Schonhage's method, or Heath-Brown's method. The complexities of these methods have exponents 1/2, 3/8, and 1/3 respectively. In this article, three new fast and potentially practical methods to compute zeta are presented. One method is very simple. Its complexity has exponent 2/5. A second method relies on this author's algorithm to compute quadratic exponential sums. Its complexity has exponent 1/3. The third method, which is our main result, employs an algorithm developed here to compute cubic exponential sums with a small cubic coecient. Its complexity has exponent 4/13 (approximately, 0.307).

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/704ad3cad6c2617cbe3f013b2ffb8c287018ea95