Is Riemann Zeta function related to differential equations?

In summary, the conversation is about a student who was surprised to see a problem involving the Riemann-zeta equations on their differential equation midterm. The study guide had mentioned to prepare for application or extension problems, but this specific problem was not covered in class or the textbook. The student questions if the Riemann-Zeta is fair game for the midterm, and another person suggests using a Fourier series to prove the result. The student expresses frustration with their professor and the difficulty of the problem.
  • #1
stgermaine
48
0
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 [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 differential equation midterm?
 
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  • #2
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
stgermaine said:
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 [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 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.
 
  • #4
stgermaine said:
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 [itex]\frac{\pi^2}{6}=1+\frac{1}{4}+\frac{1}{9}+...[/itex] seems pretty tough.
 
  • #5
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.
 

What is the Riemann Zeta function and how is it related to differential equations?

The Riemann Zeta function, denoted as ζ(z), is a mathematical function that is defined for all complex numbers except 1. It has many important applications in number theory and mathematical physics. It is also closely related to the study of prime numbers. The relationship between the Riemann Zeta function and differential equations is through the Riemann Hypothesis, which states that all non-trivial zeros of the zeta function lie on the critical line Re(z) = 1/2.

What is the Riemann Hypothesis and why is it important in relation to differential equations?

The Riemann Hypothesis is one of the most famous unsolved problems in mathematics. It was proposed by Bernhard Riemann in 1859 and states that all non-trivial zeros of the Riemann Zeta function lie on the critical line Re(z) = 1/2. It is important in relation to differential equations because a proof of the Riemann Hypothesis would have significant implications in many areas of mathematics, including number theory, analysis, and differential equations.

How does the Riemann Zeta function play a role in the study of prime numbers?

The Riemann Zeta function is closely related to the distribution of prime numbers. In particular, the Riemann Hypothesis states that the behavior of the zeta function is closely linked to the distribution of prime numbers. For example, a proof of the Riemann Hypothesis would give us a better understanding of the gaps between prime numbers and could potentially lead to more efficient methods for finding large primes.

What is the connection between the Riemann Zeta function and the Riemann integral?

The Riemann Zeta function is closely related to the Riemann integral, which is a mathematical tool for calculating the area under a curve. The Riemann integral is defined in terms of a partition of a given interval and the Riemann Zeta function is defined as a sum over all positive integers, which can be seen as a discrete version of integration. This connection allows for a deeper understanding of both concepts and has important implications in the study of calculus and analysis.

What are some real-world applications of the Riemann Zeta function and its relation to differential equations?

The Riemann Zeta function and its relation to differential equations have many real-world applications. One example is in physics, where the zeta function is used to study the behavior of quantum systems. It is also used in engineering to model the behavior of complex systems. Additionally, the Riemann Zeta function has applications in cryptography and data compression algorithms. Understanding its relation to differential equations can lead to more efficient and accurate solutions in these fields.

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