Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Explicit evaluation of Euler sums

David Borweina1, Jonathan M. Borweina2 and Roland Girgensohna3

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

Abstract

In response to a letter from Goldbach, Euler considered sums of the form

S0013091500019088_eqnU1

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

S0013091500019088_eqnU2

This evaluation involves ζ-functions and σh(2, m).

(Received August 30 1993)