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Explicit evaluation of Euler sums

Published online by Cambridge University Press:  20 January 2009

David Borwein ,
Jonathan M. Borwein  and
Roland Girgensohn
David Borwein
Affiliation:
Department of Mathematics, University of Western Ontario, London. Ontario N6A 5B7, Canada
Jonathan M. Borwein
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby. BC V5A 1S6, Canada
Roland Girgensohn
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo. Ontario N2L 3G1, Canada
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Abstract

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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).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 2 , June 1995 , pp. 277 - 294
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

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