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Riemann Zeta Function zeta(7)

zeta(7) is the value of the Riemann zeta function at 7, given by

zeta(7)=sum_(k=1)^(infty)1/(k^7)
(1)
=1.008349277381922...
(2)

(OEIS A013665).

Rapidly converging series include

zeta(7)=(19)/(56700)pi^7-2sum_(k=1)^infty1/(k^7(e^(2pik)-1))
(3)

(Plouffe 1998).

A related series is

zeta(7)=(304)/(13)sum_(n=1)^(infty)1/(n^7(e^(pin)-1))-(103)/4sum_(n=1)^(infty)1/(n^7(e^(2pin)-1))+(19)/(52)sum_(n=1)^(infty)1/(n^7(e^(4pin)-1))
(4)

(Plouffe 2006).

A binomial sum identity is

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

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.Plouffe, S. "Identities Inspired from Ramanujan Notebooks (Part 2)." Apr. 2006. http://www.lacim.uqam.ca/~plouffe/inspired2.pdf.Sloane, N. J. A. Sequence A013665 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Riemann Zeta Function zeta(7)." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RiemannZetaFunctionZeta7.html

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