### [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

## Table of Contents

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
= {*p*_{j}} where
*p*_{1} > 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*/log^{c}*x*)

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*) =
{*p*_{j}(*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
*p*_{j}(*t*) = *p*_{j}, the *j*th 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 *p*_{j}(*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*/log^{c}*x*)

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*(*x*^{1/2}) +
1/3 *P*(*x*^{1/3}) + ..., 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**.
[Contents]

**2. Generalised zeta functions**

**2.1**
Imagine the classical primes as a sequence of particles occupying
positions {*p*_{j}} 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:

[Contents]

**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 {*p*_{n}}, with energies {log *p*_{n}}. 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

*p*_{n}(*x*)=1/(*n*^{x}) = 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 *m*_{n} (*n* >
1) of the functions *p*_{n} tend slowly towards 1 as *n*.
Some examples:

*m*_{2}= 1.8791006..., *m*_{3} =
1.6351665..., *m*_{4} = 1.5329592..., *m*_{5}
= 1.4743970...

*m*_{10} = 1.3740486..., *m*_{20} =
1.3696203..., *m*_{30} = 1.3692338..., *m*_{40}
= 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 *x*1 all *p*_{n}
(*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 *p*_{n}(*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.

[Contents]

**
**## 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."

[Contents]

**
**## 7. General mathematical considerations

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

*a*_{n} /n
^{x} = *a*_{n}
exp^{i (i log n)x} = *a*_{n}[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.

[Contents]

## 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."

[Contents]

## 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/*p*_{n}}
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
.

[Contents]

## 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.

[C] P. Cohen
"Dedekind zeta functions and quantum statistical mechanics" (ESI preprint).

[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-.

[O] A. Odlyzko, "Primes,
quantum chaos, and computers"

[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.