Limit theorems for the coefficients of the modified Borwein method for the calculation of the Riemann zeta-function values

Abstract

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.