Riemann Zeta Function of 8

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Theorem

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

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


This sequence is A013666 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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.


Sources

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