An amortized-complexity method to compute the Riemann zeta function

Details An amortized-complexity method to compute the Riemann zeta function An amortized-complexity method to compute the Riemann zeta function

2010-02-11

arxiv.org/abs/1002.2443v2

Mathematics

Number Theory

13 pages, accepted in Math. Comp

Scientific paper

A practical method to compute the Riemann zeta function is presented. The method can locate all zeta zeros in [T,T+T^{1/4}] using an average time of T^{1/4+o(1)} per zero. This is the same complexity as the Odlyzko-Sch\"onhage algorithm over that interval. Although the method far from competes with the Odlyzko-Sch\"onhage algorithm over intervals much longer than T^{1/4}, it still has the advantages of being elementary, simple to implement, it does not use the fast Fourier transform or require large large amounts of storage space, and its error terms are easy to control. The method has been implemented, and results of timing experiments agree with its theoretical amortized complexity of T^{1/4+o(1)}.

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