Open Access
1994 Experimental evaluation of Euler sums
David H. Bailey, Jonathan M. Borwein, Roland Girgensohn
Experiment. Math. 3(1): 17-30 (1994).


Euler expressed certain sums of the form

\sum_{k=1}^\infty \Bigl(1 + {1 \over 2^m} + \cdots + {1 \over k^m}\Bigr) (k + 1)^{-n}\hbox{,}

where m and n are positive integers, in terms of the Riemann zeta function. In [Borwein et al.\ 1993], 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.


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David H. Bailey. Jonathan M. Borwein. Roland Girgensohn. "Experimental evaluation of Euler sums." Experiment. Math. 3 (1) 17 - 30, 1994.


Published: 1994
First available in Project Euclid: 3 September 2003

zbMATH: 0810.11076
MathSciNet: MR1302815

Primary: 11Y60

Rights: Copyright © 1994 A K Peters, Ltd.

Vol.3 • No. 1 • 1994
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