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Fast algorithms for multiple evaluations of the Riemann zeta function. (English) Zbl 0706.11047
Summary: The best previously known algorithm for evaluating the Riemann zeta function, $$\zeta (\sigma +it)$$, with $$\sigma$$ bounded and t large to moderate accuracy (within $$\pm t^{-c}$$ for some $$c>0$$, say) was based on the Riemann-Siegel formula and required on the order of $$t^{1/2}$$ operations for each value that was computed. New algorithms are presented in this paper which enable one to compute any single value of $$\zeta (\sigma +it)$$ with $$\sigma$$ fixed and $$T\leq t\leq T+T^{1/2}$$ to within $$3\pm t^{-c}$$ in $$O(t^{\epsilon})$$ operations on numbers of O(log t) bits for any $$\epsilon >0$$, for example, provided a precomputation involving $$O(T^{+\epsilon})$$ operations and $$O(T^{+\epsilon})$$ bits of storage is carried out beforehand. These algorithms lead to methods for numerically verifying the Riemann hypothesis for the first n zeros in what is expected to be $$O(n^{1+\epsilon})$$ operations (as opposed to about $$n^{3/2}$$ operations for the previous method), as well as improved algorithms for the computation of various arithmetic functions, such as $$\pi$$ (x). The new zeta function algorithms use the fast Fourier transform and a new method for the evaluation of certain rational functions. They can also be applied to the evaluation of L-functions, Epstein zeta functions, and other Dirichlet series.

MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11Y35 Analytic computations 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 65E05 General theory of numerical methods in complex analysis (potential theory, etc.) 68Q25 Analysis of algorithms and problem complexity 11Y70 Values of arithmetic functions; tables
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References:
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