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Nint Zeta Function

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Let

S_N(s)=sum_(n=1)^infty[(n^(1/N))]^(-s),
(1)

where [x] denotes nearest integer function, i.e., the integer closest to x. For s>3,

S_2(s)=2zeta(s-1)
(2)
S_3(s)=3zeta(s-2)+4^(-s)zeta(s)
(3)
S_4(s)=4zeta(s-3)+zeta(s-1).
(4)

S_N(n) is a polynomial in pi whose coefficients are algebraic numbers whenever n-N is odd. The first few values are given explicitly by

S_3(4)=(pi^2)/2+(pi^4)/(23046)
(5)
S_5(6)=(5pi^2)/6+(pi^4)/(36)+(pi^6)/(4^(12))(1/(945)-(170912+49928sqrt(2))/(25)sqrt(1-sqrt(1/2)))
(6)
S_6(7)=pi^2+(pi^4)/(18)+(pi^6)/(2520)+(246013+353664sqrt(2))/(45)(pi^7)/(2^(27)).
(7)

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References

Borwein, J. M.; Hsu, L. C.; Mabry, R.; Neu, K.; Roppert, J.; Tyler, D. B.; and de Weger, B. M. M. "Nearest Integer Zeta-Functions." Amer. Math. Monthly 101, 579-580, 1994.

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Nint Zeta Function

Cite this as:

Weisstein, Eric W. "Nint Zeta Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NintZetaFunction.html

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