Nint Zeta Function

Let

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

(1)

where

denotes Nint, the Integer closest to

. For

$\displaystyle S_2(s)$	$\textstyle =$	$\displaystyle 2\zeta(s-1)$	(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)

is a Polynomial in $\pi$ whose Coefficients are Algebraic Numbers whenever

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)

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.