# Riemann Zeta Function of 6/Proof 3

The Riemann zeta function of $6$ is given by:
 $\ds \map \zeta 6$ $=$ $\ds \dfrac 1 {1^6} + \dfrac 1 {2^6} + \dfrac 1 {3^6} + \dfrac 1 {4^6} + \cdots$ $\ds$ $=$ $\ds \dfrac {\pi^6} {945}$ $\ds$ $\approx$ $\ds 1 \cdotp 01734 \, 3 \ldots$
 $\ds \sum_{n \mathop = 1}^{\infty} \frac 1 {n^6}$ $=$ $\ds \map \zeta 6$ Definition of Riemann Zeta Function $\ds$ $=$ $\ds \paren {-1}^4 \frac {B_6 2^5 \pi^6} {6!}$ Riemann Zeta Function at Even Integers $\ds$ $=$ $\ds \frac 1 {42} \cdot \frac {2^5 \pi^6} {6!}$ Definition of Sequence of Bernoulli Numbers $\ds$ $=$ $\ds \frac {32 \pi^6} {42 \cdot 720}$ Definition of Factorial $\ds$ $=$ $\ds \frac {\pi^6} {945}$
$\blacksquare$