- #1

nomadreid

Gold Member

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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.