[Calculation 19] Even values of the Zeta Function

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Edited by Leun Kim

(revision of 2012)  In this post, we evaluate even values of the Riemann zeta function.

Calculation 19. For $n\in \N$, we have
$$
\zeta(2n) = \frac{|B_{2n}|2^{2n-1} \pi^{2n}}{(2n)!},
$$
where $B_n$’s are Bernoulli numbers.

 
Proof. We have the general formula for the generating function:
\begin{align}
\frac{z}{e^z -1} = \sum_{n=0}^\infty B_n \frac{z^n}{n!}
\end{align}
for $|z|<2\pi$, and the following identity:
\begin{align}
\frac{z}{e^z -1} + \frac{z}{2} = \frac{z}{2} \coth \frac{z}{2}.
\end{align}
Combining (1) and (2), we get
\begin{align}
z \coth z = 1 + \sum_{n=1}^\infty B_{2n} \frac{(2z)^{2n}}{(2n)!},
\end{align}
for $|z|< \pi$ where we replaced $B_0 = 1$, $B_1 = - \frac{1}{2}$, and $B_{2k+1} = 0$, $k\in \N$.   On the other hand, we have $$\sin z = z \prod_{k=1}^\infty \left( 1 - \left(\frac{z}{k\pi}\right)^2\right).$$ Taking logarithm both sides and differentiating with respect to $z$, we obtain $$ \cot z = \frac{1}{z} - 2 \sum_{k=1}^\infty \frac{z}{k^2 \pi^2 -z^2} $$ or \begin{align} z\coth z &= 1 + 2\sum_{k=1}^\infty \frac{z^2}{k^2 \pi^2 + z^2} \\ &=1 - 2 \sum_{k=1}^\infty \sum_{n=1}^\infty \left( \frac{-z^2}{k^2 \pi^2}\right)^n\\ &= 1 + 2 \sum_{n=1}^\infty \frac{\zeta(2n) (-1)^{n+1}}{\pi^{2n}} z^{2n} \end{align} for $|z|<\pi$. Comparing the coefficients of $z^{2n}$ of (3) and (6), we finally obtain $$ \zeta(2n) = \frac{(-1)^{n+1} B_{2n} 2^{2n-1} \pi^{2n}}{(2n)!} = \frac{|B_{2n}|2^{2n-1} \pi^{2n}}{(2n)!}, $$ which completes the proof.$\square$

 
References.
[1] Bruce C. Berndt, Elementary Evaluation of ζ(2n), Math. Magazine 48, No.3 (1975), 148–154.

 

 
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Leun Kim

Ph.D Candidate at The University of Tokyo
I was born and raised in Daegu, S. Korea. I majored in electronics and math in Seoul from 2007 to 2012. I've had a great interest in math since freshman year, and I studied PDE in Osaka, Japan from 2012-2014. I worked at a science museum and HUFS from 2014 in Seoul. Now I'm studying PDE in Tokyo, Japan. I also developed an interest in music, as I met a great piano teacher Oh in 2001, and joined an indie metal band in 2008. In my spare time, I enjoy various things, such as listening music, blogging, traveling, taking photos, and playing Go and Holdem. Please do not hesitate to contact me with comments, email, guestbook, and social medias.



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