Series with Binomial-Like Coefficients for Evaluation and 3D Visualization of Zeta Functions
Volume 31, Issue 4 (2020), pp. 659–680
Pub. online: 9 November 2020
Type: Research Article
Open Access
Open Access
Received
1 June 2019
1 June 2019
Accepted
1 October 2020
1 October 2020
Published
9 November 2020
9 November 2020
Abstract
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.
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