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 $\zeta(4)={\pi^4\over 90}$ .

here here

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

#Calculus #RiemannZetaFunction

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

This was an easy proof , but nicely done :)

A Former Brilliant Member - 6 years, 3 months ago

I think he forgot to mention the source . :D

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

Shivang Jindal - 6 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 $\zeta{(4)} = \dfrac{{\pi}^{4}}{90}$ .

Ronak Agarwal - 6 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 ?