Euler Sums and Contour Integral Representationsphilippe Flajolet

This paper develops an approach to the evaluation of Euler sums involving harmonic numbers either linearly or nonlinearly. We give explicit formull for certain classes of Euler sums in terms of values of the Riemann zeta function at positive integers. The approach is based on simple contour integral representations and residue computations. Sommes d'Euler et repr esentations int egrales R esum e Cet article d eveloppe une approche a l' evaluation de sommes d'Euler qui font intervenir des nombres harmoniques lin eairement ou polynomialement. Nous don-nons des formules explicites pour certaines classes de sommes d'Euler en termes de valeurs de la fonction Zeta de Riemann a des entiers positifs. Cette approche est fond ee sur des repr esentations int egrales et des calculs de r esidus. Abstract. This paper develops an approach to the evaluation of Euler sums involving harmonic numbers either linearly or nonlinearly. We give explicit formull for certain classes of Euler sums in terms of values of the Riemann zeta function at positive integers. The approach is based on simple contour integral representations and residue computations.