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# Euler Sum

In response to a letter from Goldbach, Euler considered sums of the form

 (1) (2)

with and and where is the Euler-Mascheroni constant and is the digamma function. Euler found explicit formulas in terms of the Riemann zeta function for with , and E. Au-Yeung numerically discovered

 (3)

where is the Riemann zeta function, which was subsequently rigorously proven true (Borwein and Borwein 1995). Sums involving can be re-expressed in terms of sums the form via

 (4) (5) (6)

and

 (7)

where is defined below.

Bailey et al. (1994) subsequently considered sums of the forms

 (8) (9) (10) (11) (12) (13) (14) (15)

where and have the special forms

 (16) (17) (18)

where is a generalized harmonic number.

A number of these sums can be expressed in terms of the multivariate zeta function, e.g.,

 (19)

(Bailey et al. 2006a, p. 39, sign corrected; Bailey et al. 2006b).

Special cases include

 (20)

(P. Simone, pers. comm., Aug. 30, 2004).

Analytic single or double sums over can be constructed for

 (21) (22) (23) (24) (25) (26) (27)

where is a binomial coefficient. Explicit formulas inferred using the PSLQ algorithm include

 (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42)

for ,

 (43)

for , given as a challenge problem by Borwein and Bailey (2003, pp. 24-25) and discussed in Bailey et al. (2006a, p. 39; Bailey et al. 2006b),

 (44) (45) (46)

for , and

 (47) (48)

for , where is a polylogarithm, and is the Riemann zeta function (Bailey and Plouffe 1997, Bailey et al. 1994). Of these, only (P. Simone, pers. comm., Aug. 30, 2004), , and the identities for , and have been rigorously established.

Multiple Series, Multivariate Zeta Function

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## References

Adamchik, V. "On Stirling Numbers and Euler Sums." J. Comput. Appl. Math. 79, 119-130, 1997. http://www-2.cs.cmu.edu/~adamchik/articles/stirling.htm.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2006. http://crd.lbl.gov/~dhbailey/expmath/maa-course/hyper-ema.pdf.Bailey, D. H.; Borwein, J. M.; and Girgensohn, R. "Experimental Evaluation of Euler Sums." Exper. Math. 3, 17-30, 1994.Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006.Bailey, D. and Plouffe, S. "Recognizing Numerical Constants." Organic Mathematics. Proceedings of the Workshop Held in Burnaby, BC, December 12-14, 1995 (Ed. J. Borwein, P. Borwein, L. Jörgenson, and R. Corless). Providence, RI: Amer. Math. Soc., pp. 73-88, 1997.Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, 1985.Borwein, J. and Bailey, D. "Recognition of Euler Sums." §2.5 in Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 56-58, 2003.Borwein, D. and Borwein, J. M. "On Some Intriguing Sums Involving ." Proc. Amer. Math. Soc. 123, 111-118, 1995.Borwein, D. and Borwein, J. M. "On an Intriguing Integral and Some Series Related to ." Proc. Amer. Math. Soc. 123, 1191-1198, 1995.Borwein, D.; Borwein, J. M.; and Girgensohn, R. "Explicit Evaluation of Euler Sums." Proc. Edinburgh Math. Soc. 38, 277-294, 1995.Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; and Lisonek, P. "Special Values of Multidimensional Polylogarithms." Trans. Amer. Math. Soc. 353, 907-941, 2001.Boyadzhiev, K. N. "Evaluation of Euler-Zagier Sums." Int. J. Math. Math. Sci. 27, 407-412, 2001. http://www2.onu.edu/~kboyadzh/e-zagier.pdf.Boyadzhiev, K. N. "Consecutive Evaluation of Euler Sums." Int. J. Math. Math. Sci. 29, 555-561, 2002. http://www2.onu.edu/~kboyadzh/euler-c(1).pdf.Broadhurst, D. J. "On the Enumeration of Irreducible -Fold Euler Sums and Their Roles in Knot Theory and Field Theory." April 22, 1996. http://arxiv.org/abs/hep-th/9604128Broadhurst, D. J. "Massive 3-Loop Feynman Diagrams Reducible to Primitives of Algebras of the Sixth Root of Unity." March 11, 1998. http://arxiv.org/abs/hep-th/9803091.de Doelder, P. J. "On Some Series Containing and for Certain Values of and ." J. Comp. Appl. Math. 37, 125-141, 1991.Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351-369, 1999.Flajolet, P. and Salvy, B. "Euler Sums and Contour Integral Representation." Experim. Math. 7, 15-35, 1998.

Euler Sum

## Cite this as:

Weisstein, Eric W. "Euler Sum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerSum.html