On the Evaluation of Euler Sums

Richard E. Crandall Joe P. Buhler

Abstract

Euler studied double sums of the form

ζ(r, s) = ∑_{1≤m<n} \frac{1}{n^{s}m^{r}}

for positive integers r and s, and inferred, for the special cases r = 1 or r + s odd, elegant identities involving values of the Riemann zeta function. Here we establish various series expansions of ζ(r, s) for real numbers r and s. These expansions generally involve infinitely many zeta values. The series of one type terminate for integers r and s with r + s odd, reducing in those cases to the Euler identities. Series of another type are rapidly convergent and therefore useful in numerical experiments.