Explicit evaluation of Euler sums

Abstract

In response to a letter from Goldbach, Euler considered sums of the form where s and t are positive integers. As Euler discovered by a process of extrapolation (from s + t ≦ 13), σ h ( s, t ) can be evaluated in terms of Riemann ζ-functions when s + t is odd. We provide a rigorous proof of Euler's discovery and then give analogous evaluations with proofs for corresponding alternating sums. Relatedly we give a formula for the series This evaluation involves ζ-functions and σ h (2, m ).