Summation of slowly convergent series by Gaussian type of quadratures and application to calculation of values of the Riemann zeta function

Abstract

Beside a short account on the summation/integration methods for slowly convergent series (Laplace transform method for numerical and trigonometric series and the method of contour integration over a rectangle), a method based on a new transformation of the series to a weighted integral, with respect to the weight function over R+, is presented. Such a method is applied to calculation of the values of the Riemann zeta function. Several numerical examples are included.