Waste less time on Facebook — follow Brilliant.
×

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

Note by Rajdeep Dhingra
2 years, 8 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. 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"
MathAppears 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

Sort by:

Top Newest

I think he forgot to mention the source . :D

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

Shivang Jindal - 2 years, 8 months ago

Log in to reply

Comment deleted Feb 25, 2015

Log in to reply

Please answer the reply I made to your comment.

Rajdeep Dhingra - 2 years, 8 months ago

Log in to reply

This was an easy proof , but nicely done :)

Azhaghu Roopesh M - 2 years, 8 months ago

Log in to reply

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 - 2 years, 8 months ago

Log in to reply

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

Rajdeep Dhingra - 2 years, 8 months ago

Log in to reply

I can't understand what you are trying to say here kindly clarify your statement.

Ronak Agarwal - 2 years, 8 months ago

Log in to reply

@Ronak Agarwal 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 ?

Rajdeep Dhingra - 2 years, 8 months ago

Log in to reply

@Rajdeep Dhingra You have posted a proof that doesn't use that method.

Ronak Agarwal - 2 years, 8 months ago

Log in to reply

@Ronak Agarwal I am asking just to gain knowledge not check whether my way is right. Could you prove it using that integral.

Rajdeep Dhingra - 2 years, 8 months ago

Log in to reply

@Rajdeep Dhingra Using what integral.

Ronak Agarwal - 2 years, 8 months ago

Log in to reply

@Ronak Agarwal \(\int_{0}^{\pi/2}{x^2 \ln(\cos(x))}\)

Rajdeep Dhingra - 2 years, 8 months ago

Log in to reply

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

\( cos(x) = \dfrac{e^{ix}+e^{-ix}}{2} \)

Ronak Agarwal - 2 years, 8 months ago

Log in to reply

@Ronak Agarwal NO, I meant how to prove it.

Rajdeep Dhingra - 2 years, 8 months ago

Log in to reply

@Rajdeep Dhingra 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.

Ronak Agarwal - 2 years, 8 months ago

Log in to reply

@Ronak Agarwal 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.

Rajdeep Dhingra - 2 years, 8 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...