Riemann Zeta Function zeta(9)

is the value of the Riemann zeta function at 9, given by

			(1)
			(2)

(OEIS A013667).

Rapidly converging series include

zeta(9)=(125)/(3704778)pi^9-(992)/(495)sum_(k=1)^infty1/(k^9(e^(2pik)-1))-2/(495)sum_(k=1)^infty1/(k^9(e^(2pik)+1))

(3)

(Plouffe 1998).

A related series is

(4)

(Plouffe 2006).

A binomial sum identity is

9/4sum_(k=1)^(infty)((-1)^(k+1))/(k^9(2k; k))-5/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((2)))/(k^7(2k; k))+5sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((4)))/(k^5(2k; k))+(45)/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((6)))/(k^3(2k; k))-(25)/4sum_(k=1)^(infty)((-1)^(k+1)H_(k-1)^((2))H_(k-1)^((4)))/(k^3(2k; k)),

(5)

where is a binomial coefficient and is a generalized harmonic number (Borwein and Bradley 1996, 1997; Bailey et al. 2007, p. 71).

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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

." 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.Plouffe, S. "Identities Inspired from Ramanujan Notebooks (Part 2)." Apr. 2006. http://www.lacim.uqam.ca/~plouffe/inspired2.pdf.Sloane, N. J. A. Sequence A013667 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Riemann Zeta Function zeta(9)." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RiemannZetaFunctionZeta9.html

Riemann Zeta Function zeta(9)

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