- #1

nomadreid

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## Main Question or Discussion Point

In several places, for example https://xxx.lanl.gov/pdf/chao-dyn/9406003v1, it is claimed that the Riemann zeta function is a fractal under the assumption of a positive result for the Riemann Hypothesis, because

(1) the Voronin Universality Theorem, and

(2) if the RH is true, then the zeta function fits the condition in that Theorem and hence can be arbitrarily closely approximated by an appropriate section of itself.

That is, they are implicitly using the idea of self-similarity to call it a fractal, but the definition of a fractal is not self-similarity, but rather having non-integer Hausdorff dimension. (Or at least that's one definition.) So if it is valid to call the function a fractal (under RH), then this must mean that somewhere one knows its Hausdorff dimension (or at least has appropriate bounds for it). But I haven't been able to find that. Does anyone know what that dimension is? Source?

(1) the Voronin Universality Theorem, and

(2) if the RH is true, then the zeta function fits the condition in that Theorem and hence can be arbitrarily closely approximated by an appropriate section of itself.

That is, they are implicitly using the idea of self-similarity to call it a fractal, but the definition of a fractal is not self-similarity, but rather having non-integer Hausdorff dimension. (Or at least that's one definition.) So if it is valid to call the function a fractal (under RH), then this must mean that somewhere one knows its Hausdorff dimension (or at least has appropriate bounds for it). But I haven't been able to find that. Does anyone know what that dimension is? Source?