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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: $(7)$