# Hausdorff dimension of Riemann zeta function assuming RH

• I
Gold Member

## 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?

jedishrfu
Mentor
From Wikipedia, the fractal definition using Hausdorff dimension is somewhat restrictive and that Mandelbrot himself modified the definition to include other kinds of shapes as fractals:

There is some disagreement amongst mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals."[13] More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."[14] Later, seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way."[15] Still later, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants."[16]

The consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth.[1][2][3] Fractals are not limited to geometric patterns, but can also describe processes in time.[6][4][17][18][19][20] Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds[21] and found in nature,[22][23][24][25][26] technology,[27][28][29][30] art,[31][32] architecture[33] and law.[34] Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals.[35]
https://en.wikipedia.org/wiki/Fractal

so I think the folks you're reading have taken the self similar definition approach.

Gold Member
Thanks, jedishrfu, that makes sense. So, to rephrase: does the zeta function fit Mandelbrot's earlier definition? If so, has anyone calculated the Hausdorff-Besicovich dimension?

jedishrfu
Mentor

Klystron and Matt Benesi
After I'm done laughing at Jedi's last comment. Again. And again. And again.

Can the Riemann zeta function approximate a scaled version of itself (Voronin's) despite being meromorphic (not in scope of Voronin's)?

Chris King has some zeta function fractals at his website (dhushara). Maybe ask him. He's pretty helpful when he has time. I was doing some zeta- pseudo fractals a while back (they were... sort of self similar on the same scale, so not fractal, really, and it was more Lerch).

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Klystron, jim mcnamara and jedishrfu
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Thanks for the suggestions (and pretty pics), Matt Benesi; I have a question:
Riemann zeta function [is] not in scope of Voronin's
jive with " The Riemann Hypothesis is known to be true iff ζ(s) approximates itself uniformly in the sense of Voronin's theorem (Bohr 1922, Bagchi 1987)." from http://mathworld.wolfram.com/VoroninUniversalityTheorem.html ?
I shall now try to follow up on your suggestion to contact Chris King.

Thanks for the suggestions (and pretty pics), Matt Benesi; I have a question:

jive with " The Riemann Hypothesis is known to be true iff ζ(s) approximates itself uniformly in the sense of Voronin's theorem (Bohr 1922, Bagchi 1987)." from http://mathworld.wolfram.com/VoroninUniversalityTheorem.html ?
I shall now try to follow up on your suggestion to contact Chris King.
I'd think that since the Riemann $\zeta$ has a pole at 1 it isn't a holomorphic function. Although its only pole is at 1, so...???

"Can the Riemann zeta function approximate a scaled version of itself (Voronin's) despite being meromorphic (not in scope of Voronin's <sic>theorem </sic>: it is not a holomorphic function because it has a pole at 1)?"

I'm thinking the pole prevents that. In other words, the RZ cannot accurately approximate itself because of the pole. Can you visualize a way in which it could?

Gold Member
The sense in which it is a fractal is explained by the proof of Corollary 1 in
https://arxiv.org/abs/chao-dyn/9406003
My impression (though I'm not sure, which is why I'm posting this) is that this means that certain regions of the RZ can be approximated in another region of the RZ. (The paper points out that it is not in the sense of iteration.)

Well, yeah, there is reflection too... so there are 2 T_m that approximate it. :bugeyes: Whoa.... mind blown. :D

But I still think they cannot approximate disks that include the pole?... did I miss them constraining f(s) to holomorphic parts of the zeta function? Am I supposed to assume they did?

They say "Riemann zeta function is fractal in the sense that Mandelbrot set is fractal (self-similarities between a region bounded by a closed loop C and other regions bounded by closed C′m of the same shape at smaller scales and/or at different orientations)."

I'm not too satisfied with that usage of the word fractal.... although I suppose it didn't bother me when they claimed Finnegan's Wake was the most fractal book written (on Earth Prime... <--making assumptions that I don't feel are very... true).

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RZ is not holomorphic, it is meromorphic with a discrete set of poles, so I can imagine that one could make a case for it being "almost everywhere" analytic and therefore "almost everywhere" able to be approximated. Or, in the words of Wikipedia
"The theorem as stated applies only to regions U that are contained in the strip. However, if we allow translations and scalings, we can also find encoded in the zeta functions approximate versions of all non-vanishing holomorphic functions defined on other regions. In particular, since the zeta function itself is holomorphic, versions of itself are encoded within it at different scales, the hallmark of a fractal." https://en.wikipedia.org/wiki/Zeta_function_universality

RZ is not holomorphic, it is meromorphic with a discrete set of poles, so I can imagine that one could make a case for it being "almost everywhere" analytic and therefore "almost everywhere" able to be approximated. Or, in the words of Wikipedia
"The theorem as stated applies only to regions U that are contained in the strip. However, if we allow translations and scalings, we can also find encoded in the zeta functions approximate versions of all non-vanishing holomorphic functions defined on other regions. In particular, since the zeta function itself is holomorphic, versions of itself are encoded within it at different scales, the hallmark of a fractal." https://en.wikipedia.org/wiki/Zeta_function_universality
Lol. This thread is fractal. That's a quote from the first link you posted, posted on Wikipedia.

Gold Member
Oops.
OK, I guess the only conclusion I am coming away with is that the use of the word fractal here is very loose, and since it is not strictly applicable because of the poles, one cannot technically assign it a specific Hausdorff definition.