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  • Proceedings of the Edinburgh Mathematical Society, Volume 38, Issue 2
  • June 1995, pp. 277-294

Explicit evaluation of Euler sums

  • David Borwein (a1), Jonathan M. Borwein (a2) and Roland Girgensohn (a3)
  • DOI:
  • Published online: 01 January 2009

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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1.D. H. Bailey , J. Borwein and R. Girgensohn , Experimental evaluation of Euler sums, Experiment. Math. 3 (1994), 1730.

2.B. C. Berndt , Ramanujan's Notebooks, Part I(Springer-Verlag, New York, 1985).

4.P. J. De Doelder , On some series containing Ψ(x) — Ψ(y) and (Ψ(x)—Ψ(y))2 for certain values of x and y, J. Comput. Appl. Math. 37 (1991), 125141.

6.M. Hoffman , Multiple harmonic series, Pacific J. Math. 152 (1992), 275290.

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