*(English)*Zbl 0920.11061

The authors survey some of the methods that have been used to study Euler sums, and they introduce a powerful new approach. They apply residue calculus to integrals of the form

where ${\int}_{\left(\infty \right)}$ is the limit of integrals taken along large circles that expand to $\infty $, $r\left(s\right)$ is a rational function that is $O\left({s}^{-2}\right)$ for large $\left|s\right|$, and $\xi \left(s\right)$ is a kernel function that is $o\left(s\right)$ on large circles whose radii tend to $\infty $. By employing kernels that are polynomials in $\psi \left(s\right)={{\Gamma}}^{\text{'}}\left(s\right)/{\Gamma}\left(s\right)$, its derivatives and related trigonometric functions, they deduce a host of known relations on Euler sums and discover many new ones. A modification also gives results on alternating Euler sums.

##### MSC:

11M06 | $\zeta \left(s\right)$ and $L(s,\chi )$ |