Riemann Zeta Function at Even Integers/Examples/26

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Example of Riemann Zeta Function at Even Integers

The Riemann zeta function of $26$ is given by:

\(\ds \map \zeta {26}\) \(=\) \(\ds \dfrac 1 {1^{26} } + \dfrac 1 {2^{26} } + \dfrac 1 {3^{26} } + \dfrac 1 {4^{26} } + \cdots\)
\(\ds \) \(=\) \(\ds \dfrac {\pi^{26} \times 2^{24} \times 76 \, 977 \, 927} {27!}\)


Proof

\(\ds \map \zeta {26}\) \(=\) \(\ds \paren {-1}^{14} \dfrac {B_{26} 2^{26} \pi^{26} } {26!}\) Riemann Zeta Function at Even Integers
\(\ds \) \(=\) \(\ds \dfrac {76 \, 977 \, 927} {108} \dfrac {2^{26} \pi^{26} } {26!}\) Definition of Sequence of Bernoulli Numbers
\(\ds \) \(=\) \(\ds \dfrac {\pi^{26} \times 2^{24} \times 76 \, 977 \, 927} {27!}\) simplifying

The decimal expansion can be found by an application of arithmetic.

$\blacksquare$


Sources

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