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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

() r(s)ξ(s)ds,

where () is the limit of integrals taken along large circles that expand to , r(s) is a rational function that is O(s -2 ) for large |s|, and ξ(s) is a kernel function that is o(s) on large circles whose radii tend to . By employing kernels that are polynomials in ψ(s)=Γ ' (s)/Γ(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.

11M06ζ(s) and L(s,χ)