zbMATH — the first resource for mathematics

Experimental evaluation of Euler sums. (English) Zbl 0810.11076
The authors consider eight classes of infinite series they call Euler sums. Two representative examples are \[ \sum_{k=1}^ \infty \bigl( 1+ {\textstyle {1\over 2}}+ \cdots+ {\textstyle {1\over k}} \bigr)^ m (k+1)^{-n} \quad \text{ and } \quad \sum_{k=1}^ \infty \bigl(1+ {\textstyle {1\over {2^ m}}} +\cdots+ {\textstyle {1\over {k^ m}}} \bigr) (k+1)^{-n}, \] where \(m\), \(n\) are positive integers with \(n\geq 2\); the other six are variants with alternating signs. A method is described for high-precision numerical evaluation of all these sums, and some of them are evaluated explicitly in closed form in terms of the Riemann zeta function. Many of the closed-form identities were detected experimentally. It is an open question whether such closed-form identities exist for all the Euler sums considered in this paper.

11Y60 Evaluation of number-theoretic constants
40A25 Approximation to limiting values (summation of series, etc.)
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
[1] Abramowitz M., Handbook of Mathematical Functions (1972) · Zbl 0543.33001
[2] Atkinson K. E., An Introduction to Numerical Analysis (1989) · Zbl 0718.65001
[3] Bailey D. H., ACM Transactions on Mathematical Software
[4] Bailey D. H., Math. Comp. 53 pp 649– (1989)
[5] Berndt B. C., Ramanujan’s Notebook (1985) · Zbl 0555.10001
[6] Borwein D., ”On an intriguing integral and some series related to {\(\zeta\)}(4),” · Zbl 0840.11036
[7] Borwein D., ”Explicit evaluation of Euler sums,” · Zbl 0819.40003
[8] Borwein J. M., Pi and the AGM (1987)
[9] Borwein J. M., Amer. Math. Monthly 96 pp 681– (1989) · Zbl 0711.11009
[10] DOI: 10.1007/978-1-4615-7386-9
[11] DOI: 10.1016/0377-0427(91)90112-W · Zbl 0782.33001
[12] Ferguson H. R. P., ”A polynomial time, numerically stable integer relation algorithm,”
[13] Ferguson H. R. P., Bull. Amer. Math. Soc. pp 912– (1979) · Zbl 0424.10021
[14] Ferguson H. R. P., J. Algorithms pp 131– (1987) · Zbl 0649.10021
[15] Hastad J., SIAM J. Computing 18 pp 859– (1988) · Zbl 0692.10033
[16] Knuth D. E., The Art of Computer Programming 1 (1973) · Zbl 0302.68010
[17] Knuth D. E., The Art of Computer Programming 2 (1981) · Zbl 0477.65002
[18] DOI: 10.1007/BF01457454 · Zbl 0488.12001
[19] Lewin L., Polylogarithms and associated functions (1981) · Zbl 0465.33001
[20] Wolfram S., Mathematical A System for Doing Mathematics by Computer,, 2. ed. (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.