Informatica

In this paper, the numerical algorithms for calculating the values of the left- and right-sided Riemann–Liouville fractional integrals and the Riesz fractional integral using spline interpolation techniques are derived. The linear, quadratic and three variants of cubic splines are taken into account. The estimation of errors using analytical methods are derived. We show four examples of numerical evaluation of the mentioned fractional integrals and determine the experimental rate of convergence for each derived algorithm. The high-precision calculations are executed using the 128-bit floating-point numbers and arithmetic routines.

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.