# 'prime evolution' notes

### [third version - 27/08/04]

If you were not one of the recipients of the original announcement e-mailed on 23/11/99, then it's probably a good idea to read this explanation first.

If you've already seen a version of this page, you may be interested in a new development which appears to put the main idea (1.5 below) in an entirely new light.

original version of these notes (23/11/99)

second version of these notes (23/11/99) - almost identical to this document

Warning: The contents of this document are somewhat vague. This is an unavoidable consequence of (1) the unorthodox nature of the material and (2) my relative unfamiliarity with some of the technical issues involved and general reliance on intuition. Rather than approaching this as a coherent whole, I suggest you see it as a loose collection of curious notions, amongst which there may be one or two useful new ideas. If you wish to demonstrate that this is all nonsense (and you may be right - I just don't know), then you are invited to explain why for the benefit of myself and others.

Note: When I discovered this extraordinary preprint in April 2001, I sensed that it might provide a certain framework within which to meaningfully reformulate my speculations:

I.V. Volovich, "Number theory as the ultimate physical theory", p-Adic Numbers, Ultrametric Analysis and Applications 2 (2010) 77–87

1. 'Arithmetic dynamics' and Beurling's generalised prime construction

2. Generalised zeta functions

3. More dynamics (quantisation, etc.)

4. 1/f noise and self-organising criticality

5. Partition functions and probability densities

6. Random matrix theory (the Gaussian Unitary Ensemble)

7. General mathematical considerations

8. General physics considerations

9. Other ideas for dynamics

10. Miscellaneous questions

References

## 1. 'Arithmetic dynamics' and Beurling's generalised prime construction

1.1 Suppose we have an unbounded, nondecreasing sequence = {pj} where p1 > 1. Let denote the multiplicative semigroup generated by .

1.2 The sequence of prime numbers, unlike most sequences , has the property that the elements of its multiplicative semigroup = are equally spaced.

One might guess that the primes are unique in this way, but this is not the case. For if we remove 2 from the sequence of primes to give = {3,5,7,11,13,...}, our multiplicative semigroup now consists of the set of odd integers, which of course is also equally spaced.

1.3 Suppose we now define two counting functions P(x) and N(x), which count the number of elements of and less than or equal to x, respectively. We are now able to ask certain questions about the 'stability' of results concerning the prime numbers.

For example, we know when is the sequence of primes, P(x) satisfies

P(x) ~ x/log x

(the prime number theorem), and N(x) is just the simple unit-step function [x]. One obvious question concerns the extent to which we can deform without affecting the property N(x) ~ 1. In other words, when does a sequence generate a multiplicative semigroup with 'asymptotic density' = 1?

1.4 These matters were considered by A. Beurling when he introduced his 'generalised prime construction' in 1937. He called the elements of 'generalised primes' or 'g-primes', and the elements of 'generalised integers' or 'g-integers'. In this context, the familiar prime numbers {2,3,5,7,...} are referred to as the 'classical primes', and as the 'classical integers'.

The essence of the theory is that if a sequence of g-integers is generated by a sequence of g-primes, and if one of the sequences is distributed sufficiently like its classical counterpart, then so too is the other.

To be more precise, Beurling proved

Theorem: If N(x) satisfies the asymptotic relation

N(x) = Ax + O(x/logcx)

for some A > 0 and c > 3/2, then the conclusion of the prime number theorem is valid for the system of g-primes , that is

P(x) ~ x/log x.

A comprehensive bibliography and set of notes concerning the Beurling construction can be found here.

1.5 Suppose we devise a dynamical system which involves an evolving sequence (t) = {pj(t)}, and where the evolution (t here being a 'time' parameter) is in some sense 'caused' or 'driven' by the unevenness of the spacing in its (evolving) multiplicative semigroup (t) at each time t.

The idea is that the sequence of primes represents a 'balanced' or 'equilibrium' state, an 'attractor' where the forces responsible for the evolution vanish. Through a process of self-organisation or feedback, a sequence could conceivably evolve towards 'primeness' and stop when it reaches this state (where each pj(t) = pj, the jth prime). We can think of 'primeness' in this situation as a kind of 'arithmetic equilibrium' where addition and multiplication are in harmony with each other, characterised by the geometric regularity of the multiplicative semigroup = .

[This idea preceded my awareness of Beurling's work. Many thanks to M. Huxley for bringing it to my attention after seeing the original version of these notes]

1.6 After posting the original version of these notes I became aware that B. Julia had been thinking along similar lines. In a personal communication (30/12/99) he explained:

"...the chemical potential I introduced in my 1989 lecture and discussed further in Strasbourg Dec. 1983 paper and Physica A 1994 article (with Beurling in the title) had actually been introduced by Sathe and Selberg before. This parameter gives a flow in the space of Beurling theories. I first mentioned Beurling gases in a talk at ENS Summer institute before Strasbourg. Since about 4 years [ago] I am really thinking in terms of evolution too, I should look at your notes to see if there is any overlap there. I was indeed rather busy unfortunately."

The 1994 article in question is: B. Julia, "Thermodynamic limit in number theory: Riemann-Beurling gases" Physica A 203 (1994) 425-436.

Having looked at this, I've not found any obvious overlap with my ideas here. However, I'd be interested to know what precisely Julia is refering to when he says "the space of Beurling theories". For some time I've been wondering what kind of space would be the appropriate setting for the 'evolution' of sequences described in 1.5.

1.65 [note added 27/08/04] The following preprint has just come to my attention:

T.W. Hilberdink and M.L. Lapidus, "Beurling zeta functions, generalised primes, and fractal membranes" (preprint 08/04)

The following excerpt from the introductory section refers to Lapidus's forthcoming book In Search of the Riemann Zeros, and the fact that it will include a new body of ideas associated with a

"continuous-time 'dynamical deformation' of Beurling zeta functions and prime systems [which] would provide a new away to understand the remarkable role played (within the broader class of Beurling-type zeta functions) by arithmetic (or number-theoretic) zeta functions, such as the Riemann zeta function..."

1.7 One possible scheme involves treating the evolving sequence of g-primes (t) as a set of charged particles moving in a fixed 1-dimensional field. The charges on the particles fluctuate deterministically based on the structure of (t), the associated multiplicative semigroup generated at each time t.

The charge on a particle/g-prime pj(t) would perhaps be given by an infinite sum of weighted contributions from each of its multiples in (t). Larger multiples would make proportionally smaller contributions, in such a way that these charge-sums are finite. The contributions would be based on some kind of 'density' measurable at each g-integer in . Each g-integer can be thought of as 'trying to influence' each of its g-prime factors in such a way as to appropriately increase or decrease the density at its own location. The process would continue until uniform density 1 were achieved throughout (t), i.e. when the evolving sequence (t) generating it had been forced into a state of 'primeness'.

1.8 Originally I'd thought of point-charge particles (with fixed charges) in a responsively fluctuating field, but for various reasons I now think that the idea of fluctuating charges in a fixed field is more plausible.

1.9 The idea of charged particles suggests further forces due to inter-particle interaction. If this were involved, it is unclear whether it would be inverse-square-based mutual repulsion, something logarithmic (where particles were mutually attractive at long range and mutually repulsive at short range), or something altogether different. Also there is the issue of whether it should be a 'nearest neighbour' interaction or a universal interaction.

1.10 The pole s = 1 of the Riemann zeta function might possibly correspond to some kind of singularity in the field in which the particles are moving. However, as is constructed from the classical primes (or integers), we should consider the appropriate 'modified' zeta functions for general systems of g-primes. These will be examined in section 2.

1.11 An idea suggesting an entirely different approach to constructing a dynamical system can be found in Beurling's original 1937 paper, and again in H. Diamond's 1969 article on the Beurling generalisation.

Beurling was trying demonstrate the 'sharpness' of his central theorem by constructing an example of a system of g-primes where the relation

N(x) = Ax + O(x/logcx)

is satisfied by the associated g-integers for A > 0 and c = 3/2, but the g-primes fail to satisfy the 'prime number theorem' relation P(x) ~ x/log x. Recall from 1.4 that this relation is necessarily satisfied when c > 3/2.

He succeeded in doing this, but chose to use a continuous analogue of a g-prime system. His example involved a continuous 'prime measure' and associated 'integer measure', which differ from the usual (atomic) counting measures of prime number theory.

In a 1970 article, H. Diamond produced a conventional system of (atomic) g-primes with the required property, thereby showing that the use of continuous g-prime measures is not necessary to demonstrate that Beurling's theorem [1.4] is sharp.

In a personal communication (21/09/00) Diamond stated

"The reason that Beurling used the continuous measures rather than atomic measures is that they are simpler and more natural than the others. He was such a good mathematician, that I am sure that if he had wanted to make a discrete example, he certainly could have done so. My discrete example was in fact based on Beurling's example. Roughly speaking, my prime counting function is [  ] of Beurling's counting function, where [  ] is the greatest integer function."

Continous g-prime measures appear again in Diamond's 1969 article. The idea is to replace the counting functions P(x) and N(x) with more general functions which need not be unit-step functions, and which can even be continous.

Diamond introduces increasing functions N(x) and supported in and connected by the relation

Here N(x) generalises the usual g-integer counting function, but is not the generalisation of the usual g-prime counting function P(x). Rather, it generalises P(x) + 1/2 P(x1/2) + 1/3 P(x1/3) + ..., which counts g-primes and their powers, with inverse-power weight, a phenomenon familiar from the theory of the classical prime distribution.

This suggests the possibility of an evolving continuous (rather than discrete) system of g-primes resulting, via some deterministic feedback or self-organisation process, in the classical prime distribution. This animation, although not directly related, has some suggestive value here:

We see the gradual emergence of the classical primes as pointlike objects from a continuum, here based on the nontrivial zeros of the Riemann zeta function . The image of stars condensing out of clouds of gas comes to mind. Although the animation suggests a continuous mathematical process, it is based on a set of discrete steps, one for each pair of nontrivial zeros. There are a number of ways we could transform this into a continuous process ('blurring out' the zeros of in various ways) but no one of then appears particularly obvious, natural or canonical.

[thanks to Raymond Manzoni for this animation]

1.12 Beurling's theorem stated in 1.4 may be of some significance in the construction of a dynamical system which will generate the classical primes. He has basically shown that the equal spacing of g-integers is caused by g-primes being distributed according to the logarithmic law

P(x) ~ x/log x

by which the classical primes distribute.

M. Wolf has discovered the presence of 1/f noise in the classical prime distribution. This is a property of power-frequency spectra, and has been shown to be closely related to the phenomenon of self-organised criticality by Bak, Tang and Wiesenfeld.

Wolf has shown that the 1/f noise is due to the logarithmic distribution ~ x/log x rather than the arithmetic 'primeness' of the primes, or anything related to the local fluctuations in their density. Therefore we have the following triangle of possible associations:

A final observation: If we go back to the explicit formula of Riemann and von Mangoldt, we see that the approximation x/log x for is due only to the residue contribution of the simple pole at s = 1. The effect of the nontrivial zeros of the zeta function (which govern the local fluctuations) are not taken into account. We will return to these matters in section 4.

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## 2. Generalised zeta functions

2.1 Imagine the classical primes as a sequence of particles occupying positions {pj} in a 1-dimensional field. Now imagine these particles being perturbed, or moving slightly. We can then simply adapt Euler's infinite product

to arrive at the idea of 'modified zeta functions', which evolve continuously with a continuously evolving sequence of particles (system of g-primes).

These generalisations of Riemann's zeta function have been studied in the theory surrounding Beurling's g-prime construction.

2.2 Note how a g-zeta function relates a sequence of g-primes to its multiplicative semigroup of g-integers via the analog of the usual formula

which relates the set of classical primes to its multiplicative semigroup, .

A few questions come to mind (H. Diamond's response to these can be found here):

2.3 For which sequences will the corresponding g-zeta functions have range of convergence , like the usual Riemann zeta function?

2.4 For which of these sequences will the corresponding g-zeta functions allow analytic continuation to \ {1}?

2.5 For which of these sequences will the (analytic continuations of the) g-zeta functions have simple poles at s= 1?

2.6 For which of these sequences will the corresponding zeta functions produce sets of 'trivial' zeros on the negative real axis? Sets of 'nontrivial' zeros contained within the critical strip 0 <Re s < 1 ? 'Nontrivial' zeros symmetric with respect to the critical line Re s = 1/2 ?

2.7 For which of these sequences will the corresponding g-zeta functions produce sets of nontrivial zeros with GUE-like spacing statistics ? (The usual zeta function has this property. GUE = Gaussian Unitary Ensemble.)

2.8 Consider how the prime-counting step function can be expressed as a limit of sums of smooth functions based on powers where the are the zeros of the usual Riemann zeta function. Suppose then that we 'perturb' the classical primes slightly to produce a system of g-primes . Corresponding to this we have both a g-prime counting function P(x) (a generalisation of ) as well as a g-zeta function with a new set of zeros in .

Under what conditions can we guarantee that the analogous limit function, built from powers of x where the exponents are the zeros of the g-zeta function, will coincide with P(x)?

Equivalently we can ask which systems of g-primes have valid 'explicit formulae' analogous to those of Riemann and von Mangoldt.

2.9 Corresponding to an evolving g-zeta function, we have an evolving set of zeros in . It seems possible that by making the appropriate restrictions on the evolution of the g-prime sequence, we can guarantee the 'nontrivial' zeros are in the critical strip, or perhaps even on the critical line.

Suppose the Hilbert-Polya conjecture is correct, and the complex zeros of the usual zeta function do replicate the spectrum of eigenvalues of a Hermitean operator on a Hilbert space. We might then consider in what ways we can modify the classical primes so that the resulting sets of zeros in continue to correspond to the spectra of such operators, for each time t. This leads us to the idea of an evolving operator:

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## 3. More dynamics (quantisation, etc.)

3.1 The last section concluded with the (rather nonlinear) idea of a quantum system where the operator evolves with the evolution of the state vector. Someone has almost certainly developed this idea if indeed it can be given a precise meaning. We shall proceed under the assumption that it can.

3.2 An evolving operator will give us an evolving spectrum of eigenvalues. Therefore in certain special situations, an evolving operator will correspond to an evolving g-zeta function (whose 'nontrivial' zeros match the spectrum of the operator at each time t).

3.3 The hypothesised Hilbert-Polya operator, which would prove the Riemann Hypothesis, might then be the 'final state' of the evolving operator. In this scenario, an evolving system of g-primes has reached 'primeness' or 'arithmetic harmony' as described earlier, and stopped evolving. Its evolution was accompanied by an evolving g-zeta function, which corresponded to an evolving operator, as described in 3.2. At the moment the sequence stopped, the operator was fixed at one particular 'value'. Could the Hilbert-Polya operator possibly be something like this?

3.4 This brings us to the idea of a kind of feedback loop. Although there is no one method for determining a quantisation of a classical (Hamiltonian) system, the reverse process is straightforward - replace the noncommutative operators associated with the physical observables by simple commuting variables. In this way, a feedback could possibly be introduced. The 'evolving operator' discussed above would give rise to an evolving underlying classical Hamiltonian. This could possibly 'drive' the evolution of the system of g-primes. The idea is summarised informally in the following 'causative loop':

System of g-primes evolves g-zeta function evolves set of zeros in evolves corresponding operator (with matching spectrum) evolves 'underlying' Hamiltonian evolves system of g-primes evolves [according to Hamiltonian at each moment]

Can this be made meaningful? If so could it be in some way compatible with the arithmetic dynamics suggested in Section 1?

3.5 The ideas concerning continuous prime- and integer-measures in 1.11 suggest that it may be appropriate to allow the g-prime 'particles' to (individually) be 'distributed objects' in some context. Could this might relate to some kind of quantisation of the arithmetic dynamics?

3.6 If, due to quantisation, we had probability densities rather than actual positions for the g-prime particles, the semiclassical limit should produce Dirac delta functions, in accordance with the density function .

H. Diamond's comments on continuous prime distributions can be found here.

3.7 If the imaginary parts of the nontrivial zeros are to behave like eigenvalues, we might consider what the corresponding eigenstates (base states) could be. Consider the expansion

where R is the the function introduced by Riemann to approximate , and the zeros (trivial and nontrivial) of the zeta function.

Could this identity be related to a dynamical law which holds throughout the evolutionary process?

The prime fluctuation function D(x) = - R(x) is an infinite sum of 'exponential rescalings' of Riemann's function R(x). It is quite remarkable that a single smooth function can provide not only an excellent estimate for , but also, through an infinite sum of these 'rescalings', an exact expression for the remainder (fluctuation function) - R(x).

It would seem that the base states are going to relate somehow to the various rescalings of R, or 'prime harmonics'.

A graph of the prime density fluctuation function D(x) from [GHR]. This seemingly 'noisy signal' can be decomposed into 'harmonics' corresponding directly to the zeta zeros. Michael Berry and others have often used musical analogies.

3.8 Presumably, if the arithmetic dynamics could be modeled, there would be a phase space involved. In a Hamiltonian setting, this would normally be considered as the union of constant energy surfaces, on which conservative systems evolve.

However, the arithmetic dynamics need not be conservative. In [GHR], the authors calculate the four Liapunov exponents of the prime density fluctuation function, arriving at a sum of -0.07. As there is a margin of error involved, and this sum is reasonably close to zero, the authors safely conclude that the "unknown dynamical system" associated with the prime distribution could be either conservative or dissipative.

Note that the authors were approaching the prime distribution from the point-of-view of interested chaos theorists. They applied a technique to the primes which is normally applied by physicists to experimental data generated by a (possibly unknown) dynamical system, to determine if it is chaotic, etc. It seems that they were not (consciously) suggesting that the primes are the result of some dynamical evolution, as I am here.

[Contents]

## 4. 1/f noise and self-organising criticality

4.1
In his paper [W], Marek Wolf shows that the classical primes, treated as a 'signal', display 1/f noise (this is a form of self-similarity). He shows that this is not a consequence of the arithmetic 'primeness' of the primes, but of the fact that they are distributed according to the logarithmic law

~ x/log x.

Any such sequence will display 1/f noise when considered in the same way.

This suggests something like a 'surface' L of similarly distributed systems of g-primes (in whichever space we are using).

4.2 In [BTW] Per Bak, Chao Tang, and Kurt Wiesenfeld demonstrated a link between 1/f noise and self-organised systems. Wolf cites this, and it inspired the extraordinary concluding question "Are the prime numbers in a self-organized critical state?" Although this is not meant to be more than speculation, reading it made me feel that my strange 'evolutionary' ideas might not be completely meaningless.

4.3 More information and some useful links relating to 1/f noise and self-organising critical systems can be found here.

4.4 Note that 1/f noise appears in physical systems as diverse as sunspots, quasars, hourglasses, rivers, electronic components, economies, DNA codes, speech and written language. Bak, et.al. imply that all of these are in some sense self-organised. They put forward a simple model (which has become known as the "sandpile dynamics") to suggest how this might be possible. However, an earlier model they propose in [BTW], involving a sequence of nearest-neighbour interacting torsion-pendula, may be more immediately compatible with these ideas.

4.5 It is possible that something closely related to the 'sandpile dynamics' could be directly involved in the evolutionary dynamics underlying the primes. The idea is to look at gaps between consecutive primes, and their deviations from the local average. These values can then be treated as 'heights' of a 1-dimensional sandpile at each position.

The sandpile dynamics might have to be adapted so that 'continua' rather than 'quanta' of sand flowed from one position to its neighbours, and so that the mechanism governing this was probabilistic rather than deterministic. In this way, the 1-dim. sandpile would be seen to gradually 'level itself out' or approach equilibrium, corresponding to a system of g-primes on the surface L. The 'arithmetic' forces might then be responsible for the finer points of the distribution.

4.6 One obstacle is that Bak, Tang and Wiesenfeld suggest that 1/f noise is a temporal indication of self-organising criticality, and self-similarity is the spatial indication. The 1/f noise in the primes would appear to be more of a spatial than a temporal matter. However, the ideas of 'space' and 'time' are somewhat ambiguous in the context of an evolutionary dynamics which somehow generates the prime distribution. Also, in some of the examples of 1/f noise (see Wentian Li's extensive 1/f Noise Bibliography) it's hard to see how the noise could be a temporal attribute.

This brought to mind the curious quotation from J.J. Sylvester:

"I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitations of our faculties in regard to time, which like space may be in essence poly-dimensional and that this and such sort of truths would become self-evident to a being whose mode of perception is according to superficially as opposed to our own limitation to linearly extended time."

from Collected Mathematical Papers, Volume 4, page 600

which in modern English says something like:

"I have sometimes thought that the distribution of prime numbers only seems mysterious to us because of our limitations in perception related to time. Time, like space, may be multi-dimensional, and perhaps to a being who could perceive a more generalised kind of time there would be no mystery surrounding prime numbers - it would all be obvious."

4.7 In a personal communication, Michael Berry stated that he thought the 1/f noise in the primes was "not fundamental" and only due to a "failure to rescale the primes so that their mean spacing is zero". He pointed out that once rescaled, "the fluctuations separate cleanly from the mean density, and have Poisson character (apart from fine-scale arithmetic features such as that embodied in the Hardy-Littlewood conjecture)".

With the utmost respect to Professor Berry, I feel that he might be overlooking the point of what Wolf has discovered. I can see the value of rescaling spectra of energy levels, etc. and studying their statistics. looking at the statistics of the rescaled nontrivial zeta zeros has proven similarly rewarding, and as he points out here, the very structure of the prime distribution suggests rescaling. But why dismiss a potentially crucial result due to a "failure to rescale"? The primes, in their natural state, show a very particular kind of logarithmic density, which is mathematically linked to the remarkable and ubiquitous phenomenon of 1/f noise. This may be coincidental, but it's at least worth considering, especially with all of the emerging, and largely mysterious links between primes and various aspects of modern physics.

4.8 Taking Wolf's work into account, the theorem of Beurling stated in 1.4 suggests a possible link between 1/f noise and those systems of g-primes whose g-integers are asymptotically 'equally spaced'.

Although this involves a couple of speculative leaps, we could imagine that some kind of self-organisation process is responsible for bringing about the asymptotic 'equal spacing' in the evolving g-integers.

The exact equal spacing of is ultimately linked to the local fluctuations in the distribution of primes, closely related to the nontrivial zeros of the Riemann zeta function. This suggests that some further force, beyond those responsible for the self-organisation, is responsible for the 'finer points' of the distribution of the classical primes.

Recall from 1.12 that if we go back to the explicit formula of Riemann and von Mangoldt, we see that the approximation x/log x for is due only to the residue contribution of the simple pole at s = 1. The effect of the nontrivial zeros of the zeta function (which govern the local fluctuations) are not taken into account.

4.9 Note that you can replace or delete any finite portion of any g-prime sequence in L and it will remain in L.

4.10 Within L, the sequence of primes obviously distinguishes itself. This distinction can be represented by the deviations of the primes from the asymptotic logarithmic approximation. The above diagram from [GHR] illustrates these deviations (actually the closely related - R(x) rather than - x/log x), as does the 'scattering' around the base of the curves in Figure 2 of [W] (below). The deviations are directly related to the zeros of the Riemann zeta function.

4.11 On page 368 of [BTW], the authors explain that "Once relaxed, the properties of the [self-organised critical] state are probed by locally perturbing the system." If the primes are in a self-organised critical state, as Wolf hints, then we might try to imagine the result of this kind of probing.

[Contents]

## 5. Partition functions and probability densities

5.1
In his article [J], Bernard Julia constructs the free Riemann gas, an abstract numerical 'gas' whose particles are the primes {pn}, with energies {log pn}. The construction is quite simple and uncontrived. Using fermionic statistics and considering the grand canonical ensemble, he is able to show that its partition function is the Riemann zeta function . Here the number of particles N is allowed to vary freely in the ensemble and volume V is not involved. Temperature T is the only parameter of the partition function, where x = 1/T, x being the parameter of the real-valued zeta function.

Recall that the partition function is the fundamental object of study for a system in thermodynamics, much as is the fundamental object of study in analytic number theory.

5.2 Since the seminal paper of Lee and Yang [LY], the extension of partition functions to the complex plane, and the study of their zeros there has been commonplace. Julia points out that "Lee-Yang type theorems restrict the locus of zeros of partition functions that may be related to phase transitions. The Riemann Hypothesis is similar." He also relates the pole of the Riemann zeta function at s = 1 to a phase transition known as a Hagedorn catastrophe.

5.3 As the spectrum of allowed energies over the ensemble is {log n : n = 1,2,3...} and corresponds directly to , the partition function is basically providing a different probability distribution over the natural numbers for each x in the interval . Recall that a partition function 'divides up' a unit of probability between all allowable energies, based on the proportion of the systems in the ensemble which possess each energy.

Although it is not clear to me what these probabilities ultimately refer to in the context of [J], I suspect that they might be involved in what I'm trying to achieve.

5.4 Representing the distribution over as the partition of a vertical unit into sections corresponding to the sequence of probabilities, and placing these vertical units above each x on the horizontal axis, a sequence of curves emerges, the first few illustrated below:

Equivalently, we can consider the individual function

pn(x)=1/(nx) = probability of n 'occurring' at (inverse 'temperature') x.

Again, it's not clear what the 'occurring' means here. Julia seems to be ambiguous on this point (or there is some subtlety I have missed). To understand what these probabilities actually refer to, I believe we really need a number theoretic understanding of 'temperature' in this context, as they are T -dependent.

The partition function of a system is conventionally derived from certain basic assumptions of thermodynamic theory. Trying to adapt the same derivation to the (number theoretic) free Riemann gas in order to fully understand it would be very helpful here.

We find that the locations of the maxima mn (n > 1) of the functions pn tend slowly towards 1 as n. Some examples:

m2= 1.8791006..., m3 = 1.6351665..., m4 = 1.5329592..., m5 = 1.4743970...
m10 = 1.3740486..., m20 = 1.3696203..., m30 = 1.3692338..., m40 = 1.3690822...

The areas under these curves are finite, so we could normalise and arrive at a sequence of probability densities, one for each n over the interval . It's not clear whether this would be helpful or actually refer to anything, but it's worth a mention.

If these probabilities are saying anything, it is (loosely speaking) this:

In some unknown context, n = 1 is certain to 'occur' at absolute zero temperature (T = 0, so x is infinite), and all other natural numbers n are most likely to 'occur' near 1, with larger numbers tending to 'occur' closer and closer to 1.

Note that as x1 all pn (x)0. As x gets closer to 1, the probabilities over the elements of become more and more 'equally distributed'. At x, 'total equality' is achieved at the expense of all probabilities vanishing. Nothing can 'occur' when T = x = 1. This is related to the Hagedorn catastrophe Julia mentions.

5.5 It had occured to me that these probabilities suggest proportions of an ensemble of systems of Beurling g-primes. Various schemes can be contrived to establish such correspondences, but I now find this approach to be naive.

Also note that generalised zeta functions of Section 2 will produce modifications of the 'Julia partitions' for each x. Hence the individual probability functions pn(x) described above could be seen as evolving.

5.6 The idea of probability density functions associated with individual numbers might somehow relate to the quantisation of the evolutionary dynamics (if indeed such a thing can be defined).

5.7 There have been several variations on Julia's free Riemann gas concept:

• In [J], Julia himself introduces the Möbius gas which is similar to the free Riemann gas, but involves fermionic rather than bosonic statistics. As mentioned earlier in 1.6, Julia also went on to introduce Riemann-Beurling gases in [J2].

• In a recent lecture in Budapest, Marek Wolf introduced a "prime gas" where energies are based on the gaps between primes, rather than on their actual magnitudes. In this way, his thermodynamics involves something like a volume V, which [J] fails to do. In Wolf's gas, the partition function suggests that the primes behave like noninteracting harmonic oscillators.

• In [BC] Jean-Benoit Bost and Alain Connes describe a dynamical system (in C* algebra formalism) whose partition function is , and where the pole at s = 1 corresponds to a spontaneous breaking of symmetry.

• D. Harari and E. Leichtnam generalised this to the general number field case in [HL].

• In [C], Paula Cohen improved upon [HL] by presenting a generalisation whose partition function is the Dedekind zeta function corresponding to the number field in question.

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## 6. Random matrix theory (the Gaussian Unitary Ensemble)

6.1
Much work has been done by Andrew Odlyzko, Michael Berry and others to demonstrate the similarity between the statistics of Gaussian Unitary Ensemble eigenvalue spacings and the statistics of the nontrivial zeta zero spacings.

One blatant similarity is the following pairwise correlation function which applies in both settings:

Here g(r) is the probability that you will find another eigenvalue (zero) a distance r away from a given eigenvalue (zero). This formula had been discovered in the context of random matrix theory by Freeman Dyson, and separately for zeta zero statistics by Hugh Montgomery. They coincidentally became aware of the similarity in each others' work in 1972 while talking informally over tea at Princeton where Montgomery was a visitor. A graph of g(x) looks like this:

It is often said that this suggests a 'repulsion' between zeros, unlike the Poisson spacing statistics.

If we imagine that the distribution of primes is the result of an evolutionary process, then the zeta zeros will be the result of an accompanying 'dual' evolution (see Section 2). Could this 'repulsion' evident in the 'final state' of the zeta zeros'evolution somehow reflect a repulsive tendency within the evolutionary dynamics suggested in section 1 (and/or its dual)?

6.2 The GUE is an ensemble, that is, a space of unitary matrices together with a particular probability density. Matrices in the GUE are often referred to as 'random matrices'. Although such statements can be made rigorous, there is a subtlety here that should not be overlooked. An individual matrix with fixed entries can never be 'random'. The randomness is only present in the wider context of the probability density over the entire ensemble.

The statistics discussed in 6.1 relate to the behaviour of eigenvalues of matrices taken at random from this ensemble. It is not difficult to make precise and meaningful statements about this. Now if it exists, the hypothetical Hilbert-Polya operator is one fixed operator, it cannot be 'random', yet it is often said that it behaves like an (arbitrarily large) random unitary matrix. Such statements are based on the remarkable correspondence of statistical properties discussed above.

To me, this suggests that the statistics of the zeta zeros might be an indication of some kind of (evolutionary) 'history' associated with the zeta function, and the related Hilbert-Polya operator. For the connection with the GUE suggests an entire class of operators, and that the operator we're looking for has taken other 'values' in the 'past'. The GUE probability density could then be a clue as to the actual dynamics governing the evolution. It seems to suggest that the evolving Hermitean operator is more likely to take some values than others, in a very particular way.

However, this view may be entirely misguided. These comments from [BK] (p.245) may help to clarify:

"It is important to note that we are here considering individual systems and not ensembles, so statistics cannot be defined in the usual way, as ensemble averages. Instead, we rely on the presence of an asymptotic parameter: high in the spectrum (or for large t in the Riemann case), there are many levels (or zeros) in a range where there is no secular variation, and it is this large number that enables averages to be performed. Universality then emerges in the limit h-bar 0 (or t ) for correlations between fixed numbers of levels or zeros.

A mathematical theory of universal spectral fluctuations already exists in the more conventional context where statistics are defined by averaging over an ensemble. This is random-matrix theory, where the correlations between matrix eigenvalues are calculated by averaging over ensembles of matrices whose elements are randomly distributed, in the limit where the dimension of the matrices tends ot infinity. Here the relevant ensemble is that of complex hermitean matrices: the "Gaussian unitary ensemble" (GUE). As will be discussed in the next section, it is precisely these statistics that apply to high eigenvalues of individual chaotic systems without time-reversal symmetry, and also to high Riemann zeros, in the sense that the spectral or Riemann-zero averages described in the previous paragraph coincide with GUE averages."

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## 7. General mathematical considerations

7.1
Dirichlet series are some kind of hyperbolic or logarithmic generalisation of Fourier series, in that

an /n x = an expi (i log n)x = an[cos(i log n)x + sin (i log n)x],

which could be then be expressed in terms of cosh and sinh.

This suggests that Dirichlet series are in some sense analogous to Fourier series, but concerned with periodicities on a logarithmic scale (i.e. self-similarity). I had wondered if an analogous 'Dirichlet analysis' had been developed. Julia pointed out to me recently that this is all made explicit in Hardy and Wright's book [HW].

Note that (1) the Riemann zeta function is the prototype Dirichlet series (all coefficients equal 1), and (2) the decomposition of - R(x) into 'prime harmonics' (based on the zeros of ) discussed in Section 3 is somewhat reminiscent of a Fourier decomposition, but dealing with different scaling regimes rather than with periodicities.

Also note that Marek Wolf has discovered at least three different kinds of self-similarity within the distribution of primes.

7.2 It's important to understand the role of the logarithm function in relating the continuum to the half-continuum. The log function obviously maps the half interval onto the real line in such a way that the multiplicative identity 1 gets mapped to the additive identity 0. The reason that the primes are logarithmically 'spaced out' would appear to be closely linked to the fact that the domain in which they exist has an edge (i.e. they live in a half-continuum).

It may also be of some interest to note that the graph of y = log x is one of the simplest self-similar objects which can be constructed. Its self-similarity is evident when you rescale the x-axis, and the graph simply 'scrolls' up or down parallel to the y-axis.

7.3 I suspect that following mathematical structures may also be of some use: Clifford algebras (particularly David Hestenes' closely related geometric algebras of arbitrary dimension), and p-adic numbers.

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## 8. General physics considerations

8.1
The possibility of nearest-neighbour interaction in the hypothesised 'arithmetic dynamics' described in Section 1 above brings to mind lattice gases, Ising models, and the Bak-Tang-Wiesenfeld torsion pendulum model (see [BTW]).

Also, Wolf and Julia have pointed out that the Kramers-Wannier duality [KW], which relates to Ising models and lattice gases, is strongly analogous to the functional equation of the Riemann zeta function.

8.2 For many years, Andrew Odlyzko has been developing algorithms for computing zeros of the Riemann zeta function. At the time of writing of [O], his fastest algorithm was "very similar to the one proposed by Greengard and Rokhlin [GR] for astrophysical many-body simulations, and could be used in its place."

8.3 Freeman Dyson [D] linked random matrices and Brownian motion. Patrick Billingsley [B] linked Brownian motion and prime numbers, and more recently H.Gopalkrishna Gadiyar and R. Padma [GP] have linked the Wiener-Khintchine formula to the distribution of prime pairs. This formula is normally applied in the modelling of Brownian motion (among other physical phenomena).

As discussed in [BK] random matrices are linked to the primes through the GUE spacing statistics of the nontrivial zeros of the Riemann zeta function. This is absolutely central to the 'spectral interpretation' approach to the Riemann hypothesis.

Is this 'triangle of relationship' fully understood? Could Brownian motion have some role to play in the evolutionary dynamics?

After discovering the paper "Ladder heights, Gaussian random walks, and the Riemann zeta function" by the statisticians J. Chang and Y. Peres, which involves issues related to Brownian motion, I contacted Professor Peres and asked:

"Preumably you are aware of the connections between random matrix theory (particularly the Gaussian Unitary Ensemble) and the spectral interpretation of the Riemann zeta function. It seems quite plausible that what you have done may be related. I seem to recall that Dyson demonstrated a relationship between random matrix theory and Brownian motion.

Any thoughts on this?"

He replied:

"I am aware of the classical relations you mention, but not of any explicit connection of them to our work. It would be great to find such a connection. Another relevant survey paper by Biane, Pitman and Yor is attached."

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## 9. Other ideas for dynamics

9.1
The evolutionary dynamics could perhaps be modeled in such a way that the process had a very definite starting point. It might be possible to introduce something analogous to the 'big bang', where all of the particles begin at a single position (say at 1). Some infinitesimal perturbation, together with a mutual repulsion between the particles, might then result in an 'explosion' in the positive direction. This could initiate the evolutionary process which terminates with the familiar distribution of prime numbers.

Intuition suggests that the immediate obstacles to giving this a coherent mathematical description might be overcome through the application of non-standard analysis.

Note that this 'starting point' doesn't technically correspond to a system of g-primes, as the sequence of particles is not unbounded.

9.2 Evolution might not be based on conservation of energy (as in conventional Hamiltonian mechanics) but conservation of some other quantity. If the surface L (see 4.1) does turn out to be an appropriate setting in which evolution could occur, could it be a surface on which some quantity was preserved? What, if anything, could this other quantity be? What is preserved when a system of g-primes evolves on L?

9.3 Evolution in a lattice rather than a continuum?

9.4 Evolution in rather than in +?

9.5 If we also consider the set of inverse primes {1/pn} in the interval (0,1) as part of the dynamics, then the associated multiplicative semigroup will be ideally evolving towards the positive rational numbers + rather than towards . Could some aspect of measure theory be applied to the closure in this situation, in order to define charges on our particles in a way which would work? Is there some sense in which + has a 'uniform density' but a general multiplicative semigroup does not?

9.6 The explicit formula of Riemann and von Mangoldt suggests possibly treating the x-dependent residues associated with the singularities of - as describing 'force fields' in .

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## 10. Miscellaneous questions

10.1 Has anyone looking at the primes from a thermodynamic viewpoint attempted to apply the concept of entropy?

10.2 Is there a simple way to compute the trivial zeros of a generalised zeta function? Can the evenly-spaced nature of the trivial zeros of the Riemann zeta function be seen to be directly linked to the evenly-spaced nature of the multiplicative semigroup generated by the primes (i.e. ) in the wider context of g-zeta functions? (see section 2)

10.3 Repeated application of the Riemann zeta function to the interval leaves the point 1.83377... fixed. What else does it do? Some kind of stretching/compression, presumably?

## References

[BK] M. Berry and J. Keating, "The Riemann zeros and eigenvalue asymptotics", SIAM Review 41, no. 2 (1999) 236-266.

[W] M. Wolf, "1/f noise in the distribution of prime numbers", Physica A, 241 (1997) 493-499.

[GHR] Z.Gamba, J.Hernando and L.Romanelli, "Are the prime numbers regularly ordered?", Physics Letters A 145, no.2,3 (1990) 106-108.

[HW] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford (1945)

[BTW] P.Bak, C. Tang and K. Wiesenfeld ,"Self-organized criticality", Physical Review A, 38, no.1 (1988) 364-374.

[J] B. Julia, "Statistical theory of numbers" from Number Theory and Physics (Springer Proceedings in Physics, Volume 47. editors Luck, Moussa and Waldschmidt, 1990)

[J2] B. Julia, "Thermodynamic limit in number theory: Riemann-Beurling gases", Physica A 203 (1994) 425-436.

[BC] J.-B. Bost and A. Connes "Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory," Selecta Mathematica (New Series), 1 (1995) 411-457.

[HL] D. Harari and E. Leichtnam, "Extension du phenomene de brisure spontanee de symetrie de Bost-Connes au cas des corps globaux quelconques," Selecta Mathematica, (New Series), 3 (1997) 205-243.

[LY] T. Lee and C.Yang, "Statistical theory of equations of state and phase transitions", Physical Review, 87, no.3 (1952) 404-419.

[KW] H.A.Kramer and G.H.Wanier, "Statistics of the two-dimensional ferromagnet. Part I", Physical Review 60 (1941) 252-.

[GR] L. Greengard and V. Rokhlin, "A fast algorithm for particle simulations", Journal of Computational Physics, 73 (1987) 325-348.

[D] F.J. Dyson, "A Brownian-motion model for the eigenvalues of a random matrix", Journal of Mathematical Physics, 3 (1962), 1191-1198.

[B] P. Billingsley, "Prime numbers and Brownian motion", American Mathematical Monthly, 80 (1973) 1099-1115.

[BK] M.V. Berry and J.P. Keating, "The Riemann zeros and eigenvalue asymptotics", SIAM Review, 41, No.2 (1999), 236-266.

[GP] H.Gopalkrishna Gadiyar and R.Padma, "Ramanujan-Fourier series, the Wiener-Khintchine formula and the distribution of prime pairs", Physica A 269 (1999) 503-510.