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Recently, I have been able to derive the power series for functions which contain the Bernoulli numbers, namely the $\dfrac{x}{e^x-1}$, $\cot(x)$, $\coth(x)$, $\tan(x)$, $\tanh(x)$. This leads me to consider the formula of Zeta function at even integers, which is:

$$(-1)^{n+1}\dfrac{B_{2n}2^{2n-1}\pi^{2n}}{(2n!)}$$

I find this formula very interesting because many parts of it are similar to the kernels of those trigonometric and hyperbolic functions that I mentioned above.

Looking around, I can only find one elementary "proof" of this formula in wikiproof

It is no surprise to me that the function $\pi x\cot(\pi x)$ is related to the formula by the equation:

$$\pi x\cot(\pi x)=1-2\sum_{n=1}^{+\infty}\zeta(2n)x^{2n}$$

But the proof of this formula looks kind of tricky. You add $-2$ and $-\dfrac{1}{2}$ to derive this formula.

There is another proof on wikiproof which is much longer and use Fourier analysis to derive. This is way over my head and I don't think I can understand it at this moment.

I wonder if someone can list or provide resources for a simple and elementary derivation of this recurrent formula.

I have found some papers that I cannot access, one is:

Another Proof of the Famous Formula for the Zeta Function at Positive Even Integers

Euler's formula for the zeta function at the positive even integers

If you have free access to these articles, may you mind share them with me?

James Warthington
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1 Answers1

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$$f(z)=\frac{\pi^2}{\sin^2(\pi z)}-\sum_n \frac1{(z+n)^2}$$ is a $1$-periodic thus bounded entire function, therefore it is constant: $f(z)=f(i\infty)=0$.

Next expand $\sum_n \frac1{(z+n)^2}$ in Laurent series to make $\zeta(2k)$ appear and relate $\frac{\pi^2}{\sin^2(\pi z)}$ to $\frac{z}{e^z-1}$ to make the Bernouilli numbers appear.

reuns
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