In this paper, we continue the study of efficient algorithms for the computation of zeta functions on the complex plane, extending works of Coffey, Slezeviciene and Vepstas. 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.

Numerical experiments as well as practical applications of zeta-functions require a great amount of computations. We introduce a modification of one of the efficient methods for calculating the Riemann zeta-function values Borwein's algorithm. We prove local and central limit theorems for the coefficients of a modified version of the method, as well as establish convergence rates to the limiting law. An asymptotic expression is derived for the coefficients of the modified version of the method. Calculation of the zeta-values with the resulting approximation is discussed.

The paper extends the study of applying the mixed-stable models to the analysis of large sets of high-frequency financial data. The empirical data under review are the German DAX stock index yearly log-returns series. Mixed-stable models for 29 DAX companies are constructed employing efficient parallel algorithms for the processing of long-term data series. The adequacy of the modeling is verified with the empirical characteristic function goodness-of-fit test. We propose the smart-$\Delta$ method for the calculation of the $\alpha$-stable probability density function. We study the impact of the accuracy of the computation of the probability density function and the accuracy of ML-optimization on the results of the modeling and processing time. The obtained mixed-stable parameter estimates can be used for the construction of the optimal asset portfolio.