Is Riemann Zeta function related to differential equations?

stgermaine

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?

Number Nine

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)

Mute

Quote by stgermaine

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.

micromass

Quote by stgermaine

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

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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Number Nine	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)

AlephZero $Math 2012$ Recognitions: Science Advisor	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.