Riemann Zeta Function of 8

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Theorem

$\displaystyle \map \zeta 8 = \sum_{n \mathop = 1}^{\infty} \frac 1 {n^8} = \frac {\pi^8} {9450}$

where $\zeta$ denotes the Riemann zeta function.


Proof

\(\displaystyle \sum_{n \mathop = 1}^{\infty} \frac 1 {n^8}\) \(=\) \(\displaystyle \map \zeta 8\) Definition of Riemann Zeta Function
\(\displaystyle \) \(=\) \(\displaystyle \paren {-1}^5 \frac {B_8 2^7 \pi^8} {8!}\) Riemann Zeta Function at Even Integers
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {30} \cdot \frac {2^7 \pi^8} {8!}\) Definition of Sequence of Bernoulli Numbers
\(\displaystyle \) \(=\) \(\displaystyle \frac {128 \pi^8} {30 \cdot 40 \, 320}\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle \frac {\pi^8} {9450}\)

$\blacksquare$


Historical Note

The Riemann Zeta Function of 8 was solved by Leonhard Euler, using the same technique as for the Riemann Zeta Function of 4, the Riemann Zeta Function of 4 and the Basel Problem.


Sources