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Euler sums and contour integral representations. (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 $\int_{(\infty)}r(s)\xi(s) ds,$ where $$\int_{(\infty)}$$ is the limit of integrals taken along large circles that expand to $$\infty$$, $$r(s)$$ is a rational function that is $$O(s^{-2})$$ for large $$| s|$$, and $$\xi(s)$$ is a kernel function that is $$o(s)$$ on large circles whose radii tend to $$\infty$$. By employing kernels that are polynomials in $$\psi(s)=\Gamma'(s)/\Gamma(s)$$, 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 $$\zeta (s)$$ and $$L(s, \chi)$$
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