Is Riemann Zeta function related to differential equations?


by stgermaine
Tags: differential, equations, function, riemann, zeta
stgermaine
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Dec3-12, 08:04 PM
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Hi. I just came back from my diffeq midterm and was surprised to see a problem with the Riemann-zeta equations on it. I think the problem went something like

"Prove that [itex]\pi[/itex]/6 = 1 + (1/2)^2 + (1/3)^2 + ... "

The study guide did mention that "prepare for a problem or two that may be applications or extensions of the concepts mentioned in the book". We just covered BVP, Fourier series, Wave eqn, heat eqn, and the Sturm-Liouville problems.

It doesn't show up on the textbook and I've just about had it with this prof. I write down every single thing he says in class, and it's nowhere in my notes or the textbook.

Is the Riemann-Zeta fair game for a diffeq midterm?
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Number Nine
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Dec3-12, 08:15 PM
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That's not the Riemann zeta function; the RZF is a generalization of that series on the complex plane (mind you, that particular result was one of the motivations for developing the RZF)
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Dec3-12, 08:29 PM
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Quote Quote by stgermaine View Post
Hi. I just came back from my diffeq midterm and was surprised to see a problem with the Riemann-zeta equations on it. I think the problem went something like

"Prove that [itex]\pi[/itex]/6 = 1 + (1/2)^2 + (1/3)^2 + ... "

The study guide did mention that "prepare for a problem or two that may be applications or extensions of the concepts mentioned in the book". We just covered BVP, Fourier series, Wave eqn, heat eqn, and the Sturm-Liouville problems.

It doesn't show up on the textbook and I've just about had it with this prof. I write down every single thing he says in class, and it's nowhere in my notes or the textbook.

Is the Riemann-Zeta fair game for a diffeq midterm?
One of the ways to demonstrate that sum is to derive a fourier series for a cleverly chosen function f(x), and evaluate the series at a specific value of x. It's not an uncommon example. Look at the series - can you think of a function whose fourier components give coefficients like that? You may even have derived this fourier series in class.

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Dec3-12, 08:47 PM
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Is Riemann Zeta function related to differential equations?


Quote Quote by stgermaine View Post
Is the Riemann-Zeta fair game for a diffeq midterm?
The methods that you developed in class are certainly sufficient to prove the result. But I feel that the question is a pretty difficult one if you never saw it before. I would have at least given the function of which to find the Fourier series of. Or they should have mentioned it in class. Just getting the question to prove [itex]\frac{\pi^2}{6}=1+\frac{1}{4}+\frac{1}{9}+...[/itex] seems pretty tough.
AlephZero
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Dec4-12, 06:48 AM
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If the examples in your course included Fourier series for square waves, triangle waves, etc, and you know how to differentiate and integrate Fourier series, you should be able to guess a function whose Fourier coefficients are ##1/n^2##, (and then prove your guess is correct, of course!) and use that to sum the series.


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