Riemann zeta: regularization and universality

In summary, the conversation discussed two questions related to the Riemann zeta function. The first question was about zeta function regularization and its use in calculating the value of the Riemann zeta function at -3. The second question was about the Voronin Universality function and its relation to fractals and analytic continuation. The speaker clarified that some authors refer to the zeta function as the function on the domain where the p-series converges, while others refer to the zeta function after analytic continuation. They also clarified that there is nothing inherently wrong with a fractal being smooth.
  • #1

nomadreid

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I am not sure which is the appropriate rubric to put this under, so I am putting it in General Math. If anyone wants to move it, that is fine.
Two questions, unrelated except both have to do with the Riemann zeta function (and are not about the Riemann Hypothesis).
First, in https://en.wikipedia.org/wiki/Zeta_function_regularization, it is stated:
"zeta function regularization is available appears in the Casimir effect, which is in a flat space with the bulk contributions of the quantum field in three space dimensions. In this case we must calculate the value of Riemann zeta function at -3, which diverges explicitly. However, it can be analytically continued to s=-3 where hopefully there is no pole, thus giving a finite value to the expression."
But the Riemann zeta function, I thought, is already the analytic continuation of the generalized harmonic series. For instance, https://en.wikipedia.org/wiki/Particular_values_of_Riemann_zeta_function puts the value of the Riemann zeta function of -3 at 1/120. What gives?

Second question: The Voronin Universality function states that any non-vanishing analytic function can be approximated by the Riemann zeta function in a strip defined by 1/2<Re(s)<1, and as a corollary, since the Riemann zeta function is itself analytic, this makes the function a fractal in that region. Fractals are notoriously not known for their smoothness, yet I thought that an analytic continuation was supposed to be smooth. I'm obviously got at least one concept here wrong; which one(s)?

Thanks in advance for putting me straight.
 
  • #3
I think I figured out the problems (correct me if I am wrong). For my first question: some authors refer to the zeta function as the function only on the domain where the p-series converges, some refer to the zeta function after analytic continuation. For the second question: there is nothing wrong with a fractal being smooth. In a way, a straight line is a fractal.
 

1. What is the Riemann zeta function and why is it important in mathematics?

The Riemann zeta function, denoted by ζ(s), is a mathematical function that was first introduced by Bernhard Riemann in 1859. It is defined for complex numbers s, and is closely related to the distribution of prime numbers. It is important in mathematics because it has connections to many areas such as number theory, complex analysis, and physics.

2. What is regularization in the context of the Riemann zeta function?

Regularization is a technique used in mathematics to assign a value to a divergent series or integral. In the context of the Riemann zeta function, it is used to assign a value to ζ(s) at s = 1. The standard definition of the function is only valid for s with real part greater than 1, but using regularization, we can extend its domain to include s = 1.

3. How is the Riemann zeta function related to the distribution of prime numbers?

The Riemann zeta function is closely related to the distribution of prime numbers through the Euler product formula, which expresses ζ(s) as an infinite product involving prime numbers. This connection has been used to prove important results in number theory, such as the prime number theorem and the Riemann hypothesis.

4. What is universality in the context of the Riemann zeta function?

Universality is a property of the Riemann zeta function where it exhibits similar behavior to other functions in certain regions of the complex plane. In other words, the function can be approximated by other functions in these regions. This has been observed in various mathematical and physical contexts, making the Riemann zeta function a significant object of study and research.

5. Are there any unsolved problems related to the Riemann zeta function?

Yes, there are several unsolved problems related to the Riemann zeta function, the most famous of which is the Riemann hypothesis. This hypothesis, first proposed by Bernhard Riemann in 1859, states that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2. This problem has been a subject of much research and is still unsolved, with a million-dollar prize offered by the Clay Mathematics Institute for its solution.

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