# Riemann Zeta Function of 8

## Theorem

The Riemann zeta function of $8$ is given by:

 $\ds \map \zeta 8$ $=$ $\ds \dfrac 1 {1^8} + \dfrac 1 {2^8} + \dfrac 1 {3^8} + \dfrac 1 {4^8} + \cdots$ $\ds$ $=$ $\ds \dfrac {\pi^8} {9450}$ $\ds$ $\approx$ $\ds 1 \cdotp 00408 \, 3 \ldots$

## Proof

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

$\blacksquare$

## Historical Note

The Riemann Zeta Function of 8 was solved by Leonhard Euler, using the same technique as for the Riemann Zeta Function of 6, the Riemann Zeta Function of 4 and the Basel Problem.