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Fast Algorithms for Multiple Evaluations of the Riemann Zeta Function

A. M. Odlyzko and A. Schönhage
Transactions of the American Mathematical Society
Vol. 309, No. 2 (Oct., 1988), pp. 797-809
DOI: 10.2307/2000939
Stable URL: http://www.jstor.org/stable/2000939
Page Count: 13
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Fast Algorithms for Multiple Evaluations of the Riemann Zeta Function
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Abstract

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 $\pm t^{-c}$ in $O(t^\varepsilon)$ operations on numbers of $O(\log t)$ bits for any $\varepsilon > 0$, for example, provided a precomputation involving $O(T^{1/2+\varepsilon})$ operations and $O(T^{1/2+\epsilon})$ 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 $O(n^{1+\varepsilon})$ 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.

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