# Riemann Zeta Function of 8

## 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.