Riemann Zeta Function of 6

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Theorem

$\displaystyle \map \zeta 6 = \sum_{n \mathop = 1}^{\infty} \frac 1 {n^6} = \frac {\pi^6} {945}$

where $\zeta$ denotes the Riemann zeta function.


Proof

\(\displaystyle \sum_{n \mathop = 1}^{\infty} \frac 1 {n^6}\) \(=\) \(\displaystyle \map \zeta 6\) Definition of Riemann Zeta Function
\(\displaystyle \) \(=\) \(\displaystyle \paren {-1}^4 \frac {B_6 2^5 \pi^6} {6!}\) Riemann Zeta Function at Even Integers
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {42} \cdot \frac {2^5 \pi^6} {6!}\) Definition of Sequence of Bernoulli Numbers
\(\displaystyle \) \(=\) \(\displaystyle \frac {32 \pi^6} {42 \cdot 720}\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle \frac {\pi^6} {945}\)

$\blacksquare$


Historical Note

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


If only my brother were alive now.
-- Johann Bernoulli


Sources