Abstract
We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function ζ(s, a) for \(s, a \in \mathbb {C}\), along with an arbitrary number of derivatives with respect to s, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.
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Johansson, F. Rigorous high-precision computation of the Hurwitz zeta function and its derivatives. Numer Algor 69, 253–270 (2015). https://doi.org/10.1007/s11075-014-9893-1
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DOI: https://doi.org/10.1007/s11075-014-9893-1
Keywords
- Hurwitz zeta function
- Riemann zeta function
- Arbitrary-precision arithmetic
- Rigorous numerical evaluation
- Fast polynomial arithmetic
- Power series