Riemann Zeta Function

$\zeta (s)=\sum _{n=1}^{\infty }\frac{1}{n^s}=\frac{1}{1^s}+\frac{1}{2^s}+\dots$

Common Values

$s$	Value
$1$	$\infty$ (Harmonic Series)
$2$	$\frac{{\pi }^2}{6}\approx 1.6449340$ (Basel Problem)
$3$	$\approx 1.2020569$ (Apery’s Constant
$4$	$\frac{{\pi }^4}{90}\approx 1.0823232$
$5$	$\approx 1.0369277$
$6$	$\frac{{\pi }^6}{945}\approx 1.0173430$

$\zeta (s)\cong \sum _{n=1}^M\frac{1}{n^s}+{\int }_{M+\frac{1}{2}}^{\infty }\frac{1}{n^s}\mathrm{d}n$

• Approximation from KK