Riemann Zeta Function
#Math
$\displaystyle \zeta(s)=\sum_{n = 1}^{\infty} \frac{1}{n^{s}}=\frac{1}{1^{s}}+\frac{1}{2^{s}}+\ldots$
Common Values
| $\displaystyle s$ | Value |
|---|---|
| $\displaystyle 1$ | $\displaystyle \infty$ (Harmonic Series) |
| $\displaystyle 2$ | $\displaystyle \frac{\pi ^{2}}{6}\approx 1.6449340$ (Basel Problem) |
| $\displaystyle 3$ | $\approx1.2020569$ (Apery's Constant |
| $\displaystyle 4$ | $\displaystyle \frac{\pi^{4}}{90}\approx 1.0823232$ |
| $\displaystyle 5$ | $\displaystyle \approx 1.0369277$ |
| $\displaystyle 6$ | $\displaystyle \frac{\pi^{6}}{945}\approx 1.0173430$ |
$\displaystyle \zeta(s)\cong\sum_{n = 1}^{M} \frac{1}{n^{s}}+\int_{M+\frac{1}{2}}^{\infty} \frac{1}{n^{s}} , \mathrm{d}n$
- Approximation from KK