Riemann Zeta Function at Even Integers/Examples

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Examples of Riemann Zeta Function at Even Integers

Riemann Zeta Function of $2$

The Riemann zeta function of $2$ is given by:

\(\ds \map \zeta 2\) \(=\) \(\ds \dfrac 1 {1^2} + \dfrac 1 {2^2} + \dfrac 1 {3^2} + \dfrac 1 {4^2} + \cdots\)
\(\ds \) \(=\) \(\ds \dfrac {\pi^2} 6\)
\(\ds \) \(\approx\) \(\ds 1 \cdotp 64493 \, 4066 \ldots\)


Riemann Zeta Function of $4$

The Riemann zeta function of $4$ is given by:

\(\ds \map \zeta 4\) \(=\) \(\ds \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots\)
\(\ds \) \(=\) \(\ds \dfrac {\pi^4} {90}\)
\(\ds \) \(\approx\) \(\ds 1 \cdotp 08232 \, 3 \ldots\)


Riemann Zeta Function of $6$

The Riemann zeta function of $6$ is given by:

\(\ds \map \zeta 6\) \(=\) \(\ds \dfrac 1 {1^6} + \dfrac 1 {2^6} + \dfrac 1 {3^6} + \dfrac 1 {4^6} + \cdots\)
\(\ds \) \(=\) \(\ds \dfrac {\pi^6} {945}\)
\(\ds \) \(\approx\) \(\ds 1 \cdotp 01734 \, 3 \ldots\)


Riemann Zeta Function of $8$

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

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


Riemann Zeta Function of $26$

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

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