In this paper, we continue the study of efficient algorithms for the computation of zeta functions on the complex plane, extending works of Coffey, Šleževičienė and Vepštas. We prove a central limit theorem for the coefficients of the series with binomial-like coefficients used for evaluation of the Riemann zeta function and establish the rate of convergence to the limiting distribution. An asymptotic expression is derived for the coefficients of the series. We discuss the computational complexity and numerical aspects of the implementation of the algorithm. In the last part of the paper we present our results on 3D visualizations of zeta functions, based on series with binomial-like coefficients. 3D visualizations illustrate underlying structures of surfaces and 3D curves associated with zeta functions. PDF  XML

中文翻译：

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具有二项式系数的系列用于Zeta函数的评估和3D可视化

在本文中，我们继续研究在复杂平面上计算zeta函数的有效算法，扩展了Coffey，Šleževičienė和Vepštas的著作。我们证明了具有二项式系数的级数系数的中心极限定理，该系数用于评估黎曼zeta函数，并建立了收敛到极限分布的速率。为该系列的系数导出一个渐近表达式。我们讨论了算法实现的计算复杂度和数值方面。在本文的最后一部分中，我们基于zeta函数的3D可视化展示了基于二项式系数的序列的结果。3D可视化显示了与zeta函数关联的表面和3D曲线的基础结构。PDF XML