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# Proof of $$\zeta(4) = \frac{\pi^4}{90}$$

Consider the function $$f(t):=t^2\ \ (-\pi\leq t\leq \pi)$$, extended to all of $${\mathbb R}$$ periodically with period $$2\pi$$. Developping $$f$$ into a Fourier series we get $t^2 ={\pi^2\over3}+\sum_{k=1}^\infty {4(-1)^k\over k^2}\cos(kt)\qquad(-\pi\leq t\leq \pi).$ If we put $$t:=\pi$$ here we easily find $$\zeta(2)={\pi^2\over6}$$. For $$\zeta(4)$$ we use Parseval's formula $\|f\|^2=\sum_{k=-\infty}^\infty |c_k|^2$ . Here $\|f\|^2={1\over2\pi}\int_{-\pi}^\pi t^4\>dt={\pi^4\over5}$ and the $$c_k$$ are the complex Fourier coefficients of $$f$$. Therefore $$c_0={\pi^2\over3}$$ and $$|c_{\pm k}|^2={1\over4}a_k^2={4\over k^4}$$ $$\ (k\geq1)$$. Putting it all together gives $$\zeta(4)={\pi^4\over 90}$$.

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Source - http://math.stackexchange.com/questions/28329/nice-proofs-of-zeta4-pi4-90

Note by Rajdeep Dhingra
2 years, 3 months ago

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I think he forgot to mention the source . :D

http://math.stackexchange.com/questions/28329/nice-proofs-of-zeta4-pi4-90 · 2 years, 3 months ago

Comment deleted Feb 25, 2015

This was an easy proof , but nicely done :) · 2 years, 3 months ago

Remember You just recently posted a solution to my question. I don't know whether you realize or not that it contained a proof that $$\zeta{(4)} = \dfrac{{\pi}^{4}}{90}$$. · 2 years, 3 months ago

Yes , You are right . But without showing the result of that integral using a different way we can't prove it. Can we ? · 2 years, 3 months ago

I can't understand what you are trying to say here kindly clarify your statement. · 2 years, 3 months ago

I meant without proving the result of that integral in an another way we can't prove $$\zeta(4) = \frac{\pi^4}{90}$$. Can We ? · 2 years, 3 months ago

You have posted a proof that doesn't use that method. · 2 years, 3 months ago

I am asking just to gain knowledge not check whether my way is right. Could you prove it using that integral. · 2 years, 3 months ago

Using what integral. · 2 years, 3 months ago

$$\int_{0}^{\pi/2}{x^2 \ln(\cos(x))}$$ · 2 years, 3 months ago

You mean I have to prove the result $$\zeta(4) = \dfrac{{\pi}^{4}}{90}$$ from this integral without using :

$$cos(x) = \dfrac{e^{ix}+e^{-ix}}{2}$$ · 2 years, 3 months ago

NO, I meant how to prove it. · 2 years, 3 months ago

You have posted a solution for solving that integral, in your solution observe that the imaginary part of the integral is zero and you will observe that the result gets proved. · 2 years, 3 months ago

Fine , But when I solved the question by putting $$\zeta(4) = \frac{\pi^4}{90}$$ then I can't possibly use the answer to prove this.First , we need to solve that question without using the fact that $$\zeta(4) = \frac{\pi^4}{90}$$ then only are proof for it will be valid. · 2 years, 3 months ago