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Riemann Zeta Function zeta(11)

zeta(11) is the value of the Riemann zeta function at 11, given by

zeta(11)=sum_(k=1)^(infty)1/(k^(11))
(1)
=1.000494188604119...
(2)

(OEIS A013669).

Rapidly converging series include

zeta(11)=(1453)/(425675250)pi^(11)-2sum_(k=1)^infty1/(k^(11)(e^(2pik)-1))
(3)

(Plouffe 1998).

A binomial sum identity is

zeta(11)=5/2sum_(k=1)^(infty)((-1)^(k+1))/(k^(11)(2k; k))+(25)/2sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((4)))/(k^7(2k; k))-(75)/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((8)))/(k^3(2k; k))+(125)/4sum_(k=1)^(infty)((-1)^(k+1)[H_(k-1)^((4))]^2)/(k^3(2k; k)),
(4)

where (2k; k) is a binomial coefficient and H_n^((r))=sum_(j=1)^(n)j^(-r) is a generalized harmonic number (Borwein and Bradley 1996, 1997; Bailey et al. 2007, p. 71).

See also

Riemann Zeta Function

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References

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, 2007.Borwein, J. M. and Bradley, D. M. "Searching Symbolically for Apéry-Like Formulae for Values of the Riemann Zeta Function." ACM SIGSAM Bull. Algebraic Sym. Manip. 30, 2-7, 1996.Borwein, J. M. and Bradley, D. M. "Empirically Determined Apéry-Like Formulae for zeta(4n+3)." Exp. Math. 6, 181-194, 1997.Plouffe, S. "Identities Inspired from Ramanujan Notebooks II." Jul. 21, 1998. http://www.lacim.uqam.ca/~plouffe/identities.html.Sloane, N. J. A. Sequence A013669 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Riemann Zeta Function zeta(11)." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RiemannZetaFunctionZeta11.html

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