Fast and Accurate Methods for Calculating Riemann Zeta Function رسالة ماجستير

Abstract

The Riemann Zeta function z (s) occurs as an important tool in many fields of mathematics and physics. Because of the lack of the closed formulas for calculating this function, many fast and accurate approximation methods are developed and numerically verified. Direct summation is used to evaluate z (s). Results show that decreasing s increases the error for fixed number of terms. To avoid the obstacle of slowing down of convergence in the case of small values of s, some fast algorithms are used and compared to direct calculations. Due to its high efficiency, Borwein algorithm is deeply investigated and widely used in this thesis. Various classes of polynomials are incorporated in the algorithm. Results are compared. Functions of the form p(x) = xn(1􀀀x)n and the Gegenbauer functions give best approximations