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I Riemann zeta: regularization and universality

  1. Apr 19, 2016 #1


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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.
  2. jcsd
  3. Apr 25, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
  4. Apr 30, 2016 #3


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    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.
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