Optimized multiseries hypergeometric formulas for single and multiple logarithms, found via integer programming on a lattice, enable efficient high-precision computation with demonstrated results exceeding 10^11 digits and an application to 2*10^12 digits of log(10).
Rigorous high-precision computation of the Hurwitz zeta function and its derivatives
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abstract
We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function $\zeta(s,a)$ 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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UNVERDICTED 2representative citing papers
Authors set up a closed system of equations for perturbed Li-Keiper coefficients around the Koebe function and report numerical evidence that fluctuations lambda-tiny(n) remain bounded by gamma times n.
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Fast Ramanujan--type Series for Logarithms. Part II
Optimized multiseries hypergeometric formulas for single and multiple logarithms, found via integer programming on a lattice, enable efficient high-precision computation with demonstrated results exceeding 10^11 digits and an application to 2*10^12 digits of log(10).
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Analysis of a Complex approximation to the Li-Keiper coefficients around the K Function
Authors set up a closed system of equations for perturbed Li-Keiper coefficients around the Koebe function and report numerical evidence that fluctuations lambda-tiny(n) remain bounded by gamma times n.