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Explicit evaluation of Euler sums

  • David Borwein (a1), Jonathan M. Borwein (a2) and Roland Girgensohn (a3)
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), 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).

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Copyright
References
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1.Bailey, D. H., Borwein, J. and Girgensohn, R., Experimental evaluation of Euler sums, Experiment. Math. 3 (1994), 1730.
2.Berndt, B. C., Ramanujan's Notebooks, Part I(Springer-Verlag, New York, 1985).
3.Borwein, D., Borwein, J. M., On an intriguing integral and some series related to ζ(4), Proc. Amer. Math. Soc., to appear.
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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