Riemann Zeta Function at Even Integers/Examples/26

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 $

$=$

$\ds \dfrac {\pi^{26} \times 2^{24} \times 76 \, 977 \, 927} {27!}$

$\ds \map \zeta {26}$

$=$

$\ds \paren {-1}^{14} \dfrac {B_{26} 2^{26} \pi^{26} } {26!}$

$\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.

$\blacksquare$

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$