Is Riemann Zeta function related to differential equations?by stgermaine Tags: differential, equations, function, riemann, zeta 

#1
Dec312, 08:04 PM

P: 48

Hi. I just came back from my diffeq midterm and was surprised to see a problem with the Riemannzeta 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 SturmLiouville 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 RiemannZeta fair game for a diffeq midterm? 



#2
Dec312, 08:15 PM

P: 771

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)




#3
Dec312, 08:29 PM

HW Helper
P: 1,391





#4
Dec312, 08:47 PM

Mentor
P: 16,703

Is Riemann Zeta function related to differential equations? 



#5
Dec412, 06:48 AM

Engineering
Sci Advisor
HW Helper
Thanks
P: 6,388

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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