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Nint Zeta Function


S_N(s)=\sum_{n=1}^\infty [(n^{1/N})]^{-s},
\end{displaymath} (1)

where $[x]$ denotes Nint, the Integer closest to $x$. For $s>3$,
$\displaystyle S_2(s)$ $\textstyle =$ $\displaystyle 2\zeta(s-1)$ (2)
$\displaystyle S_3(s)$ $\textstyle =$ $\displaystyle 3\zeta(s-2)+4^{-s}\zeta(s)$ (3)
$\displaystyle S_4(s)$ $\textstyle =$ $\displaystyle 4\zeta(s-3)+\zeta(s-1).$ (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

$\displaystyle S_3(4)$ $\textstyle =$ $\displaystyle {\pi^2\over 2}+{\pi^4\over 23046}$ (5)
$\displaystyle S_5(6)$ $\textstyle =$ $\displaystyle {5\pi^2\over 6}+{\pi^4\over 36}+{\pi^6\over 4^{12}}\left({{1\over 945}-{170912+49928\sqrt{2}\over 25} \sqrt{1-\sqrt{1\over 2}}\,\,}\right)$ (6)
$\displaystyle S_6(7)$ $\textstyle =$ $\displaystyle \pi^2+{\pi^4\over 18}+{\pi^6\over 2520}+{246013+353664\sqrt{2}\over 45} {\pi^7\over 2^{27}}.$ (7)


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.

© 1996-9 Eric W. Weisstein