Experimental Mathematics

Experimental evaluation of Euler sums

David H. Bailey, Jonathan M. Borwein, and Roland Girgensohn

Abstract

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.

Article information

Source
Experiment. Math. Volume 3, Issue 1 (1994), 17-30.

Dates
First available in Project Euclid: 3 September 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1062621000

Mathematical Reviews number (MathSciNet)
MR1302815

Zentralblatt MATH identifier
0810.11076

Subjects
Primary: 11Y60: Evaluation of constants

Citation

Bailey, David H.; Borwein, Jonathan M.; Girgensohn, Roland. Experimental evaluation of Euler sums. Experiment. Math. 3 (1994), no. 1, 17--30.https://projecteuclid.org/euclid.em/1062621000


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