Is Riemann Zeta function related to differential equations?

[stgermaine]

Hi. I just came back from my differential equation midterm and was surprised to see a problem with the Riemann-zeta equations on it. I think the problem went something like

"Prove that $\pi$/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 differential equation 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]

Hi. I just came back from my differential equation midterm and was surprised to see a problem with the Riemann-zeta equations on it. I think the problem went something like

"Prove that $\pi$/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 differential equation 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]

Is the Riemann-Zeta fair game for a differential equation 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 $\frac{\pi^2}{6}=1+\frac{1}{4}+\frac{1}{9}+...$ 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.