Experimental Evaluation of Euler Sums

Bailey, David H. and Borwein, Jonathan M. and Girgensohn, Roland (1994) Experimental Evaluation of Euler Sums. Experimental Mathematics, 3 (1). pp. 17-30.

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Postscript
In response to a letter from Goldbach, Euler considered sums of the form \begin{eqnarray*} \sum_{k=1}^\infty \left(1 + {1 \over 2^m} + \cdots + {1 \over k^m}\right) \, (k + 1)^{-n}. \end{eqnarray*} for positive integers $m$ and $n$. Euler was able to give explicit values for certain of these sums in terms of the Riemann zeta function. In a recent companion paper, Euler's results were extended to a significantly larger class of sums of this type, including sums with alternating signs. This research was facilitated by numerical computations using an algorithm that can determine, with high confidence, whether or not a particular numerical value can be expressed as a rational linear combination of several given constants. The present paper presents the numerical techniques used in these computations and lists many of the experimental results that have been obtained.