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Abstract

The Riemann zeta function, Ϛ(s) with complex argument s, is a widely used special function in mathematics. This thesis is motivated by the need of a cost reducing algorithm for the computation of Ϛ (s) using its Euler-Maclaurin series. The difficulty lies in finding small upper bounds, call them n and k, for the two sums in the Euler-Maclaurin series of Ϛ (s) which will compute Ϛ (s) to within any given accuracy for any complex argument s, and provide optimal computational cost in the use of the Euler-Maclaurin series. This work is based on Backlund’s remainder estimate for the Euler-Maclaurin remain- der, since it provides a close enough relationship between n, k, s, and е. We assumed that the cost of computing the Bernoulli numbers, which appear in the series, is fixed, and briefly discuss how this may influence high precision calculation. Based on our study of the behavior of Backlund’s remainder estimate, we define the ‘best’ pair (n, k), and present a reliable method of computing the best pair. Furthermore, based on a compu- tational analysis, we conjecture that there is a relationship between n and k which does not depend on s. We present two algorithms, one based on our method and the other on the conjecture, and compare their costs of finding n and k as well as computing the Euler-Maclaurin series with an algorithm presented by Cohen and Olivier. We conclude that our algorithm reduces the cost of computing Ϛ(s) drastically, and that good numerical techniques need to be applied to our method and conjecture for finding n and k in order to keep this computational cost low as well.