From Fourier Series to Rapidly Convergent Series for Zeta(3)

SummaryExact values of the Riemann zeta function ζ(s) for even values of s can be determined from Fourier series for periodic versions of even power functions, but there is not an analogous method for determining exact values of ζ(s) for odd values of s. After giving a brief historical overview of ζ(s) for integer values of s greater than one, and showing how we can determine ζ(2) and ζ(4) from Fourier series, we consider the Fourier series for a continuous and piecewise differentiable odd periodic function from which we can find a series with logarithmic terms for ζ(3). Using power series for logarithmic functions on this series, a rapidly convergent series for ζ(3) is obtained. Using a partial sum of this series we can compute ζ(3) with an error which is much smaller than the error obtained by using a similar partial sum of the infinite series defining ζ(3).