On the evaluation of Euler sums

Abstract

Euler studied double sums of the form $$ \z(r,s)=\sum_{1\le m<n}{1\over n^sm^r} $$ 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 $\z(r, s)$ 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.