Riemann Zeta Function at Even Integers/Examples/2
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Example of Riemann Zeta Function at Even Integers
The Riemann zeta function of $2$ is given by:
\(\ds \map \zeta 2\) | \(=\) | \(\ds \dfrac 1 {1^2} + \dfrac 1 {2^2} + \dfrac 1 {3^2} + \dfrac 1 {4^2} + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\pi^2} 6\) | ||||||||||||
\(\ds \) | \(\approx\) | \(\ds 1 \cdotp 64493 \, 4066 \ldots\) |
This sequence is A013661 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
\(\ds \zeta \left({2}\right)\) | \(=\) | \(\ds \left({-1}\right)^2 \dfrac {B_2 2^1 \pi^2} {2!}\) | Riemann Zeta Function at Even Integers | |||||||||||
\(\ds \) | \(=\) | \(\ds \left({-1}\right)^2 \left({\dfrac 1 6}\right) \dfrac {2^1 \pi^2} {2!}\) | Definition of Sequence of Bernoulli Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \left({\dfrac 1 6}\right) \left({\dfrac 2 2}\right) \pi^2\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\pi^2} 6\) | simplifying |
$\blacksquare$
The decimal expansion can be found by an application of arithmetic.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 644 \, 934 \, 066 \ldots$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: $(7)$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 64493 \, 4066 \ldots$