a1 Department of Mathematics, University of Western Ontario, London. Ontario N6A 5B7, Canada
a2 Department of Mathematics and Statistics, Simon Fraser University, Burnaby. BC V5A 1S6, Canada
a3 Department of Pure Mathematics, University of Waterloo, Waterloo. Ontario N2L 3G1, Canada
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), σh(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 σh(2, m).
(Received August 30 1993)