## June 14, 2013

### Quasicrystals and the Riemann Hypothesis

#### Posted by John Baez Freeman Dyson is a famous physicist who has also dabbled in number theory quite productively. If some random dude said the Riemann Hypothesis was connected to quasicrystals, I’d probably dismiss him as a crank. But when Dyson says this, it’s a lot more interesting. So I’ve been trying to understand his remarks on this. And it’s been productive, in that I’ve learned some interesting things, and I now feel closer to seeing why the Riemann Hypothesis is a natural and important conjecture.

But still, I could use a lot of help: I don’t have much time for number theory, and a few pointers from experts could keep me from going down dead ends. Those of you who were using the internet around 1990 may remember newsgroups like sci.math, sci.physics, sci.math.research and sci.physics.research. That’s how internet-savvy mathematicians and physicists communicated back then. And if you were around then, you may remember Matt McIrvin, who wrote consistently intelligent and good-natured posts about physics.

I lost track of him for a while, but met him again on Google+, where has been using the open-source math software called Sage to play around with ideas from number theory.

Sage comes equipped with a table of nontrivial zeros of the Riemann zeta function computed by Andrew Odlyzko. Remember, this function is given by

$\zeta(s) = \sum_{n = 1}^\infty n^{-s}$

when $Re(s) > 1$, but it can be analytically continued over the whole complex plane except for a pole at 1, and then it turns out that $\zeta(z) = 0$ for a lot of points on a line:

$z = \frac{1}{2} + i k$

where $k$ is a real number. Here are the first few:

$k_1 \approx 14.1347$

$k_2 \approx 21.0220$

$k_3 \approx 25.0109$

The Riemann Hypothesis claims that all the zeros of the zeta function lie on this line, except for the so-called trivial zeros at $-2, -4, -6, -8$ and so on. Riemann only checked this for the first three cases… but by now people have checked it for the first 10,000,000,000,000 cases. So it seems to be true, but nobody can prove it.

The nontrivial zeros of the Riemann zeta function are really interesting. There’s no simple formula for them, but they encode information about prime numbers. Riemann was the first to notice this… but Matt ran into it on his own.

He took the first ten thousand positive numbers $k_j$ that make

$\zeta(\frac{1}{2} + i k_j) = 0,$

and he added up a bunch of functions like this:

$exp(i k_j x)$

one for each $j = 1, \dots, 10,000$.

The result is a function of $x$, and he graphed its absolute value. The graph has lots of sharp spikes! And where are these spikes? He zoomed in on the first few: He wrote:

A closeup of those first spikes. I wonder what those numbers are, exactly? Probably they’re in the literature somewhere.

I looked around and soon found that those spikes should be here:

$\ln(2), \ln(3), \ln(4), \ln(5), \ln(7), \ln(8), \ln(9), \ln(11), \ln(13), \ln(16), ...$

See the pattern?

### Dyson’s remarks

In math jargon, what Matt did is take the Fourier transform of a sum of Dirac deltas supported at the imaginary parts of the nontrivial Riemann zeta zeros. The answer seemed to be another sum of Dirac deltas, times different numbers: the different spikes in the pictures above seem to have different heights. It’s unusual to take the Fourier transform of such a spiky function and get another spiky function. And according to Freeman Dyson, this is the defining feature of a quasicrystal!

When I was looking around for clues, one of the first things I ran into was a lecture by Dyson. He never actually delivered this lecture—it was cancelled at the last minute for some reason—but it was printed here:

• Freeman Dyson, Frogs and birds, Notices of the American Mathematical Society 56 (2009), 212–223.

It was about two styles of doing mathematics, hence the curious title. He said:

The proof of the Riemann Hypothesis is a worthy goal, and it is not for us to ask whether we can reach it. I will give you some hints describing how it might be achieved. Here I will be giving voice to the mathematician that I was fifty years ago before I became a physicist. I will talk first about the Riemann Hypothesis and then about quasicrystals.

There were until recently two supreme unsolved problems in the world of pure mathematics, the proof of Fermat’s Last Theorem and the proof of the Riemann Hypothesis. Twelve years ago, my Princeton colleague Andrew Wiles polished off Fermat’s Last Theorem, and only the Riemann Hypothesis remains. Wiles’ proof of the Fermat Theorem was not just a technical stunt. It required the discovery and exploration of a new field of mathematical ideas, far wider and more consequential than the Fermat Theorem itself. It is likely that any proof of the Riemann Hypothesis will likewise lead to a deeper understanding of many diverse areas of mathematics and perhaps of physics too. Riemann’s zeta-function, and other zeta-functions similar to it, appear ubiquitously in number theory, in the theory of dynamical systems, in geometry, in function theory, and in physics. The zeta-function stands at a junction where paths lead in many directions. A proof of the hypothesis will illuminate all the connections. Like every serious student of pure mathematics, when I was young I had dreams of proving the Riemann Hypothesis. I had some vague ideas that I thought might lead to a proof. In recent years, after the discovery of quasicrystals, my ideas became a little less vague. I offer them here for the consideration of any young mathematician who has ambitions to win a Fields Medal.

Quasicrystals can exist in spaces of one, two, or three dimensions. From the point of view of physics, the three-dimensional quasicrystals are the most interesting, since they inhabit our three-dimensional world and can be studied experimentally. From the point of view of a mathematician, one-dimensional quasicrystals are much more interesting than two-dimensional or three-dimensional quasicrystals because they exist in far greater variety. The mathematical definition of a quasicrystal is as follows. A quasicrystal is a distribution of discrete point masses whose Fourier transform is a distribution of discrete point frequencies. Or to say it more briefly, a quasicrystal is a pure point distribution that has a pure point spectrum. This definition includes as a special case the ordinary crystals, which are periodic distributions with periodic spectra.

Excluding the ordinary crystals, quasicrystals in three dimensions come in very limited variety, all of them associated with the icosahedral group. The two-dimensional quasicrystals are more numerous, roughly one distinct type associated with each regular polygon in a plane. The two-dimensional quasicrystal with pentagonal symmetry is the famous Penrose tiling of the plane. Finally, the one-dimensional quasicrystals have a far richer structure since they are not tied to any rotational symmetries. So far as I know, no complete enumeration of one-dimensional quasicrystals exists. It is known that a unique quasicrystal exists corresponding to every Pisot–Vijayaraghavan number or PV number. A PV number is a real algebraic integer, a root of a polynomial equation with integer coefficients, such that all the other roots have absolute value less than one . The set of all PV numbers is infinite and has a remarkable topological structure. The set of all one-dimensional quasicrystals has a structure at least as rich as the set of all PV numbers and probably much richer. We do not know for sure, but it is likely that a huge universe of one-dimensional quasicrystals not associated with PV numbers is waiting to be discovered.

Here comes the connection of the one-dimensional quasicrystals with the Riemann Hypothesis. If the Riemann Hypothesis is true, then the zeros of the zeta-function form a one-dimensional quasicrystal according to the definition. They constitute a distribution of point masses on a straight line, and their Fourier transform is likewise a distribution of point masses, one at each of the logarithms of ordinary prime numbers and prime-power numbers. My friend Andrew Odlyzko has published a beautiful computer calculation of the Fourier transform of the zeta-function zeros . The calculation shows precisely the expected structure of the Fourier transform, with a sharp discontinuity at every logarithm of a prime or prime-power number and nowhere else.

My suggestion is the following. Let us pretend that we do not know that the Riemann Hypothesis is true. Let us tackle the problem from the other end. Let us try to obtain a complete enumeration and classification of one-dimensional quasicrystals. That is to say, we enumerate and classify all point distributions that have a discrete point spectrum […] We shall then find the well-known quasicrystals associated with PV numbers, and also a whole universe of other quasicrystals, known and unknown. Among the multitude of other quasicrystals we search for one corresponding to the Riemann zeta-function and one corresponding to each of the other zeta-functions that resemble the Riemann zeta-function. Suppose that we find one of the quasicrystals in our enumeration with properties that identify it with the zeros of the Riemann zeta-function. Then we have proved the Riemann Hypothesis and we can wait for the telephone call announcing the award of the Fields Medal.

These are of course idle dreams. The problem of classifying one-dimensional quasicrystals is horrendously difficult, probably at least as difficult as the problems that Andrew Wiles took seven years to explore. But if we take a Baconian point of view, the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible. The classification of quasicrystals is a worthy goal, and might even turn out to be achievable.

 M. J. Bertin et al., Pisot and Salem Numbers, Birkhäuser, Boston, 1992.

 A. M. Odlyzko, Primes, quantum chaos and computers, in Number Theory: Proceedings of a Symposium, 4 May 1989, Washington, DC, USA (National Research Council, 1990), pp. 35–46.

### Questions

I wanted to understand this better, so I asked around on Mathoverflow. I got a lot of help. For starters, I got pointed to some critical remarks by Nick S., who said:

Well his definition of quasicrystals is not the one used by the quasicrystal community (we actually don’t have a formal mathematical definition yet)…. His statement about icosahedral group is false, actually most 3-dimensional models don’t have any symmetry group. Same issue for the 2 dimensional quasicrystals, most of them are not related to polygons in the plane. […] I really have no idea what he mean by “it is well known that a unique quasicrystal exists corresponding to every PV number”. The existence is true, the uniqueness is far for true… Unless I’m making a terrible mistake, there are constructions which produce pure point diffractive sets from PV numbers, and they produce uncountably many… In many situations, but not always, one can probably get that most of them are “equivalent” in some sense, but not all of them… And the big issue is that any equivalence in this sense, unless one adds very strong extra conditions, allows for .. small translations of the points… And there are of course uncountably many models which are not associated to PV numbers. Another issue is that the zeroes of the RZF are not a Delone set, so anything done so far by the quasicrystal community is not relevant to the problem… And last, I really don’t see how one can go around the following issue: Let $\Lambda$ be the set of zeroes. Let $\Lambda'$ be the set obtained by moving all the zeroes, such that the $n$th zero is moved by at most $1/n$. Then diffraction cannot differentiate between $\Lambda$ and $Lambda'$.

I don’t know if these remarks are true, so if any of you know, please tell me… preferably with references!

I don’t care too much if Dyson is using ‘quasicrystal’ in a nonstandard sense. He at least seems to be hinting at a fairly precise definition, perhaps “a countable sum of Dirac deltas on $\mathbb{R}^n$ that defines a tempered distribution whose Fourier transform is a countable linear combination of Dirac deltas”. The phrase ‘defines a tempered distribution’ just means the Dirac deltas don’t bunch up too fast, so the Fourier transform is well-defined. Allowing the original sum to also be a more general linear combination of Dirac deltas might be be nice, too: then the Fourier transform of a quasicrystal would be another quasicrystal!

Anyway, what I’d like to learn is what’s known about such entities. In what sense, if any, does any Pisot–Vijayaraghavan number give a unique quasicrystal in 1 dimension? Do de Bruijn’s quasiperiodic tilings, like this one drawn by Greg Egan’s software, give quasicrystals in Dyson’s sense? Is it really true that all quasicrystals in 3 dimensions are related to the icosahedral group? What’s the theorem there? And what about the 4-dimensional pattern built from the E8 lattice—is that a quasicrystal in Dyson’s sense? Is it hard to find higher-dimensional quasicrystals, or easy?

### The Guinand–Weil explicit formula

But now I’d like to come to my actual point, which concerns this remark:

If the Riemann Hypothesis is true, then the zeros of the zeta-function form a one-dimensional quasicrystal according to the definition. They constitute a distribution of point masses on a straight line, and their Fourier transform is likewise a distribution of point masses, one at each of the logarithms of ordinary prime numbers and prime-power numbers.

Is this actually true? In other words, does the Riemann Hypothesis actually imply this? And if so, why?

At first I thought it was true. That would make a very nice bumper sticker explaining the ‘meaning’ of the Riemann Hypothesis… something like

RIEMANN ZETA ZEROS ARE FOURIER DUAL TO LOGS OF PRIME POWERS!

This is the first version of the Riemann Hypothesis I’ve seen that makes me really want it to be true. You can see it discussed in this excellent book draft here, with lots of pretty pictures:

• Barry Mazur and Richard Stein, Primes.

But it seems the truth is a bit more complicated. The truth is called the explicit formula of Guinand and Weil and it involves terms, not only for the nontrivial zeros of the zeta function, but also for the trivial zeros, and the pole. And in fact it’s best to think of this formula not in terms of the original Riemann zeta function, but the ‘corrected’ version that takes the ‘prime at infinity’ into account using the gamma function, namely:

$\Lambda(s) = \pi^{-s/2} \Gamma(s/2) \zeta(s)$

This ‘corrected’ version has the all-important symmetry:

$\Lambda(s) = \Lambda(1-s).$

that lets you see the zeros of this function, and thus all the nontrivial zeros of the original Riemann zeta function, lie in the strip $0 \le Re(s) \le 1$.

So, I still have hope for getting a conceptually clear statement of the Riemann Hypothesis that’s exactly correct! However, so far, I can’t seem to say something correct without it looking rather messy. For example, on Mathoverflow Brad Rogers stated a version of the Guinand–Weil explicit formula that looks about like this:

For a compactly supported smooth function $g : \mathbb{R} \to \mathbb{C}$ with Fourier transform $\widehat{g}$,

$\sum_k \widehat{g}(k/2\pi) = \int_\mathbb{R} [g(x)+g(-x)] e^{-x/2}d(e^x-\psi(e^x)) + \int_\mathbb{R} \frac{\Omega(\xi)}{2\pi}\hat{g}(\xi/2\pi) d\xi$

Here the sum is over $k$ such that $1/2+i k$ is a non-trivial zero of the Riemann zeta function. $\psi$ is the Chebyshev prime counting function:

$\psi(x) = \sum_{p^k\leq x} \ln(p)$

and

$\Omega(\xi) = \tfrac{1}{2}\tfrac{\Gamma\,\prime}{\Gamma}(1/4+i\xi/2) + \tfrac{1}{2}\tfrac{\Gamma\,\prime}{\Gamma}(1/4-i\xi/2) - \log \pi.$

Alas, this formula doesn’t look very ‘conceptual’, though I think I’m beginning to understand it, and Marc Palm gave a nice sketch of a proof.

In this formula, $k$ can be complex if the Riemann Hypothesis is false. But if it’s true, $k$ is always real, and the left-hand side of the big equation

$\sum_k \widehat{g}(k/2\pi)$

is really just the test function $g$ integrated against a sum of complex exponentials, one for each nontrivial zero of the zeta function. (I should warn you that these zeros come in complex conjugate pairs, so for each positive real $k$ we get a corresponding negative $k$).

The right-hand side of the big equation contains a ‘nice’ term that’s a sum over prime powers, but also some ‘corrections’ that seem to make Dyson’s claim fail to be literally true. For example, besides a linear combination of Dirac deltas at logarithms of prime powers, there’s the correction term proportional to the function $\Omega$. Owen Maresh has plotted this function: So, I’m thinking that the slow rise at right here: might not be an artifact of numerical approximations, but an actual real thing: this function $\Omega$.

So: what’s the really neat way to write an ‘explicit formula’ relating prime powers and Riemann zeta zeros… which simplifies in some way iff the Riemann Hypothesis holds? And is there a way to modify Freeman Dyson’s claim here, that makes it correct while still maintaining a connection to quasicrystals?

If the Riemann Hypothesis is true, then the zeros of the zeta-function form a one-dimensional quasicrystal according to the definition. They constitute a distribution of point masses on a straight line, and their Fourier transform is likewise a distribution of point masses, one at each of the logarithms of ordinary prime numbers and prime-power numbers. Posted at June 14, 2013 9:41 PM UTC

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### Re: Quasicrystals and the Riemann Hypothesis

The remark Dyson makes on definitions sounds rather akin to the decision by the International Union of Crystalographers to define “crystal” for the purposes of their art as “any solid having an essentially discrete diffraction diagram”.

Of course there is no need for mathematicians to adopt the same technical terminology as an International Union of Whichever (especially when “crystal” already means something very different in differential geometry), but whether you call them “quasicrystals” or “fourier-dual discretions” the notion is clearly interesting and probably worth giving some name.

Posted by: Jesse C. McKeown on June 16, 2013 6:05 AM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

Some trivial examples of nonperiodic discretely-supported measures with discretely-supported Fourier transforms:

since the fourier transform of any lattice-counting measure is essentially a dual lattice, and since fourier transforms are linear, so any sum (union) of any finite number of lattices has discrete fourier transform, whether the union be lattice or not. Here’s a deliberately provocative example: This one is provably not Delone; it’s a sum of an obvious five transforms of one lattice chosen to make the transforms overlap nontrivially.

Posted by: Jesse C. McKeown on June 16, 2013 10:39 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

Nice! So, suppose we temporarily define a quasicrystal to be a countable set of points $S \subset \mathbb{R}^n$ such that the Fourier transform of the sum of Dirac deltas at those points is a countable linear combination of Dirac deltas.

You’re pointing out that if $S$ and $T$ are disjoint quasicrystals, so is $S \cup T$. This makes me want to classify indecomposable quasicrystals $U$, meaning those such that if $U = S \cup T$ for disjoinf quasicrystals $S$ and $T$, then either $S$ or $T$ is empty.

But wait!

Puzzle 1. Show that the lattice $\mathbb{Z}^n \subset \mathbb{R}^n$ is a quasicrystal, but not indecomposable.

Puzzle 2. Show that there exists an infinite decescending sequence of quasicrystals

$\cdots \subset A_3 \subset A_2 \subset A_1 \subset \mathbb{R}^n$

none of which are indecomposable.

So we need to be a bit more clever…

Posted by: John Baez on June 17, 2013 12:20 AM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

“Nice! So, suppose we temporarily define a quasicrystal to be a countable set of points S⊂ℝ n such that the Fourier transform of the sum of Dirac deltas at those points is a countable linear combination of Dirac deltas. “

Keep in mind that since atoms cannot come arbitrarily close, we typically ask for models to be Delone set.

Also, why call it quasicrystal? Just because it sounds fancy? Using a “temporary” definition in math is not a good idea.

Read Section 3 in the Lagarias article I mentioned, he studies this type of problem there, but for Delone sets. He calls these sets “almost periodic”, because there is a strong connection between this requirement and almost periodicity. He is only interested in the case when the fourier transform is not “too big”.

Posted by: Nick S on June 21, 2013 12:37 AM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

Nick S. wrote:

Also, why call it quasicrystal? Just because it sounds fancy?

No: because that’s what Dyson did, and I’m talking about what he said, and nobody told me any more standard term for the concept I was talking about.

Read Section 3 in the Lagarias article I mentioned, he studies this type of problem there, but for Delone sets. He calls these sets “almost periodic”…

Okay, that’s good in the Delone case then. The Riemann zeta zeroes are not a Delone set… but I’ve convinced myself by now that, despite what Dyson seemed to claim, the Fourier transform of a sum of Dirac deltas supported at nontrivial Riemann zeta zeros is not a measure supported on a discrete set.

Posted by: John Baez on June 21, 2013 2:04 AM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

True, but the Cordoba results Lagarias uses only required the set being uniformly discrete, not Delone. And the zeroes are uniformly discrete.

Posted by: Nick S on June 21, 2013 7:19 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

I answer: there are many nontrivial self-covers of the torus $\mathbb{T}^2$; so I was thinking that there has to be something to do with the stack of elliptic curves, or a line fibration over it (to remember scale), and its homology…

On the other hand, because the Fourier transform is about a bilinear pairing $\langle x, p\rangle$, taking sparser lattices in position space should be dual to taking finer, or at least more-narrowly-spaced lattices in momentum space… except that the sum-of-lattices construction insists on having a more carefully-worded expression.

But in any case, $\mathbb{Z}^n$ is an $L$ such that, for $L' \lt L$, contrariwise $\hat L' \gt \hat L$, or the supports are such. The deliberately-provocative example (or one suitably perturbed) has not this property. Maybe that’ll do?

Posted by: Jesse C. McKeown on June 17, 2013 1:06 AM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

but do remind me (everyone!) to be more careful: I’ve only verified the contravariance described for sublattices $L' \lt \mathbb{Z}$, not for arbitrary subsets…

Posted by: Jesse C. McKeown on June 17, 2013 1:10 AM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

I think that the IUC committee which decided the new definition contained some mathematicians. I’m not 100% sure but I think Marjorie Senechal, Michael Baake and others were members of that committee. And I think it was actually the mathematicians which pushed towards a vague definition.

As far as I see, most people which are interested in the study of quasicrystals, actually avoid using the term quasicrystal, at least until we understand them better.

The problem of which Dirac combs have discrete Fourier transform is studied a little by Jeff Lagarias in Mathematical Quasicrystals and the Problem of Diffraction.

For Delone sets the question is settled by two Theorems of Cordoba, and I think that relatively denseness is not needed.

BTW: there is a mistake in Lagarias’ paper, he claims that the union of translates of lattices cannot be uniformly discrete unless we are in the fully periodic case. But this is only true in $\mathbb{R}^1$, in higher dimensions the union of translates of lattices can be uniformly discrete even if the lattices are not fully commensurable.

Posted by: Nick S on June 21, 2013 12:27 AM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

This last paragraph is precisely the sort of thing the Selected Papers Network is good for!

Posted by: David Roberts on June 21, 2013 12:57 AM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

You should look at work of Alain Connes ~2000.

Posted by: Prof. David Edwards on June 16, 2013 1:35 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

What I like the most about this quasicrystal stuff is that the notion of “duality” is in the air as well. There are connections with so many subjects that it has to be true, if not at all, in the overall picture. I am intrigued by the relation between crystals and quasicrystals. As far as I know, there is no book or review talking about higher-dimensional quasicrystalography (help!) with D=d+1, and d>2. It also touches the time (quasi)crystal hypothesis by Wilczek…And I believe there is another “missing gap” there. Primes are everywhere, so time quasicrystals…What are they?

The triality connection for the New Physics yet to come:

Energy-Momentum = Geometry = Numbers!

Posted by: Juan Francisco on June 16, 2013 3:49 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

Just a question for you, John… You speculate that there are some set of “point masses” associated to the Riemann zeroes. The question is:

Riemann zeroes are masses/energies of “what object/s”? If the Dyson conjecture (can we call it so) that identifies the structure of Riemann zeroes as points of some quasicrystal, and if your idea that Riemann zeroes are some kind of “masses”…Have you thought what can they be? There are some nice papers by Arefeva et al. and Dragovich about “zeta strings” that are precisely in that line… I have thought since long ago that the critical line shares many features with Regge trajectories but I have never been able to relate Riemann zeroes with the particle spectrum (and I have tried, but not hard enough I guess…).

Of course, my above comments are completely speculative… And your post about these delta function ideas have impressed me! I had not read anything like that! Personally, I believe that the right way to classify quasicrystals is via some crystal physics connection. Grossly:

Quasicrystals = Crystals in higher D projected into a lower dimension.

If this idea is also true, what is the right higher dimension whose dimension allow us to classify unidimensional quasicrystals? I think it has some sense, there are some recents works by Sierra et al. , Lapidus and Srednicki about different physical ideas behind the RH. And recently I have read a nice paper relating electromagnetism and the zeta function…http://arxiv.org/pdf/1305.2613.pdf

I recently found this paper fascinating

http://arxiv.org/pdf/1305.3759.pdf

Cheers…

Posted by: Juan Francisco on June 16, 2013 7:01 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

Juan writes:

Just a question for you, John… You speculate that there are some set of “point masses” associated to the Riemann zeroes.

I’m not “speculating”; this is just a mathematical fact. In math, a point mass is another name for a Dirac delta measure, that is, a measure assigning measure 1 to a single point in a measure space, and measure zero to all measurable sets not containing that point. Matt McIrvin was drawing, and Freeman Dyson was discussing, the Fourier transform of the measure that’s the sum of Dirac delta measures located at the imaginary parts of all the nontrivial Riemann zeta zeros.

Posted by: John Baez on June 16, 2013 7:36 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

I agree with the fact you can associate “a point mass”(a “measure”) to a Dirac delta function (measure/distribution). I am a (theoretical) physicist with a deep passion with mathematics but…You could also do it with “point charges”, …Why masses and not “charges”?That is why I recall your mathematical statement as “speculation” (perhaps without precision). Of course, maybe is only a semantical issue in mathematics, but it is important from the viewpoint of physics:

1) Point masses can be represented by delta functions (I agree!This is not an speculation)

2) Point charges can also be represented by delta functions. (Do you agree with me?)

The question is not “a minor speculation” (perhaps I was unfortunated to choose the word “speculate”, since I am not a native English speaker, as you know…I think). If there is some kind of (deep) physics behind the RH and the RZ function, to understand if the zeroes are “masses” or “charges” of some kind IS important. Perhaps we can evade the issue partially since distributions are “flexible” enough to be interpreted in different ways…

The Riemann zeroes represent some kind of “resonances” we can interpret as “energy/mass/charge” somehow…And I am not sure if mass/energy should be better understood as some kind of generalized “(gauge) charges” (that is why I cited the paper about the electrostatic connection above). I discussed this question long ago with a friend…And I have tried (without any success untill now) to relate it with some kind of “gauge” theory where “big random matrices” arise quite naturally (think about the large N expansion in QCD, for instance). I did not obtain any valuable result so, I gave up the idea until the recent advances in QCD phase transitions and the color-glass condensates and the revival of the “world crystal” approach to differnt areas of physics and mathematics I am interested in. So, I am motivated again to purse some “physmatical” approach to the RH, supported with some additional (“new”) mathematics I am studying now (tropical math and polylogarithms; I will attend a workshop on polylogs and QFT in a few months)…

Posted by: Juan Francisco on June 16, 2013 8:34 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

Juan wrote:

You could also do it with “point charges”, …Why masses and not “charges”?

‘Point mass’ is just the standard mathematical terminology here.

1) Point masses can be represented by delta functions (I agree! This is not an speculation.)

2) Point charges can also be represented by delta functions. (Do you agree with me?)

Sure. But I wasn’t talking about either of those physical concepts. Feel free to attach physical significance to what I was saying, and see if it gets you anywhere. I wasn’t. I was raising a bunch of rather technical questions about Freeman Dyson’s claims about quasicrystals and the Fourier transform of the sum of delta functions supported at the nontrivial Riemann zeta zeros. I’m still hoping a number theorist or specialist on quasicrystals will read this post and answer some those questions. Maybe they don’t work on weekends.

Posted by: John Baez on June 16, 2013 8:45 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

I’m not an expert in chemistry, but quasicrystals interest me in health science. Compared to crystals, they lack rigidity, so their dynamics must take you out beyond periodic structure into stochasticity or chaos. The problem is then that to define such a boundary for a class of materials like quasicrystals is to grasp quantitative universality, and that, notoriously, was never pinned down, through the whole Pao Alto stream of chaos research.

Symptomatic is now the term “criticality”: its sounds like the intrinsic quality of a critical point, but its actually a bleeding sore thumb of our ignorance, at this point.

Hegel in his Logic argued that the infinitessimals of Lagrange are properly qualities of matter, not quantities at all. I mention this, because he’s back in the game through Richard Brandom’s recent Locke lectures, rediscovered as the original intuitionist! A Hegel concept is like an animal: it is not to be judged true or false, but found in a domain or habitat comprising what is not inconsistent… and on that note the axiom of choice issue intervenes!

So I find your range of topics very relevant and interesting, but math still a world away from physical stuff.

Posted by: Orwin O'Dowd on August 20, 2013 4:30 AM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

I solve RIemann hypothesis

http://vixra.org/abs/1403.0184

Posted by: tai on April 8, 2015 2:16 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

Posted by: gmail.com on May 18, 2015 8:24 AM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

Hi John, let me address your final question as another question. I am assuming the Riemann hypothesis.

So if we remove Dyson’s assertion that the Fourier transform of the Riemann zeta function are $\delta$’s located at primes and powers of primes but still retain his definition of a quasicrystal, wouldn’t his main conjecture hold?

Put another way: despite the Guinand-Weil explicit formula containing the $\Omega$ terms in addition to the “nice” terms, the Fourier transform of $\zeta(s)$ still seems to be a point spectrum from the penultimate plot you show. Yes?

Posted by: Vipin on June 10, 2016 12:44 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

A couple of us wrote a paper on this in Physical Review B where we defined a lattice called the Riemann lattice based on this proposal: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.94.214204

We do not pretend to solve any mathematical problem in this work, but instead define a new stat-mech-system based on the Riemann zeros and Dyson’s proposal, which we find to display intriguing (electronic) properties.

Posted by: Vipin on December 27, 2017 5:32 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

in general there are explicit formulae for many arithmetical functions so if the riemann zeros are quasicrytasl there are several quasicrystals each one for any arithmetical function

https://vixra.org/abs/1310.0048 pages 6 to 10 in my paper

guinand-weil formula can be generalized to include möbius function , Liouville function and Euler Phi

Posted by: jose on October 23, 2018 3:37 PM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

The spikes in the blown up graph seem to be at the same abscissa locations as those of the non-zero values for the von Mangoldt arithmetic function in terms of which the logarithmic derivative of the zeta function may be simply expressed. And, as Wikipedia claims, “The Fourier transform of the von Mangoldt function gives a spectrum with spikes at abscissas equal to imaginary part of the Riemann zeta function zeros. This is sometimes called a duality.”

Posted by: Thomas C Copeland on September 20, 2019 1:57 AM | Permalink | Reply to this

### Re: Quasicrystals and the Riemann Hypothesis

I meant to say the locations of the spikes seem to correspond to the non-zero values of the Mangoldt function of the integers, which is log(n) at n a prime or a power of a prime and zero otherwise.

Posted by: Thomas C Copeland on September 20, 2019 2:33 AM | Permalink | Reply to this

### Edward

According to Moody, pure point diffractivity is not THE defining feature. In fact, there is no agreed upon definition for a mathematical quasicrystal. There are, however, several features which are representative of the phenomenon of quasiperiodicity. One of them is pure point diffractivity. Others are, for example, finite local complexity, aperiodicity, extensiveness, discreteness. The most popular definition for a mathematical quasicrystal is just any cut-and-project set. Such sets have all of Moody’s properties. An example of a cut-and-project set is that pretty Penrose Tiling at the start of this awesome post.

Posted by: Edward Voskanian on April 30, 2021 1:06 AM | Permalink | Reply to this

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