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 \(L(s, \chi)\)
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[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) · doi:10.1090/S0025-5718-1989-0979934-9
[5] Berndt B. C., Ramanujan’s Notebook (1985) · Zbl 0555.10001 · doi:10.1007/978-1-4612-1088-7
[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 · doi:10.1017/S0013091500019088
[8] Borwein J. M., Pi and the AGM (1987)
[9] Borwein J. M., Amer. Math. Monthly 96 pp 681– (1989) · Zbl 0711.11009 · doi:10.2307/2324715
[10] DOI: 10.1007/978-1-4615-7386-9 · doi:10.1007/978-1-4615-7386-9
[11] DOI: 10.1016/0377-0427(91)90112-W · Zbl 0782.33001 · doi:10.1016/0377-0427(91)90112-W
[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 · doi:10.1090/S0273-0979-1979-14691-3
[14] Ferguson H. R. P., J. Algorithms pp 131– (1987) · Zbl 0649.10021 · doi:10.1016/0196-6774(87)90033-2
[15] Hastad J., SIAM J. Computing 18 pp 859– (1988) · Zbl 0692.10033 · doi:10.1137/0218059
[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 · doi:10.1007/BF01457454
[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)
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