EXPLICIT EVALUATION OF EULER SUMS

In response to a letter from Goldbach, Euler considered sums of the form [GRAPHICS] where s and t are positive integers. As Euler discovered by a process of extrapolation (from s+t less than or equal to 13), sigma(h)(s,t) can be evaluated in terms of Riemann zeta-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 [GRAPHICS] This evaluation involves zeta-functions and sigma(h)(2,m).