The central idea here concerns the distribution of prime numbers. It came out of dream-consciousness late in 1998 in a vague but powerfully memorable form. It is possible that the idea is entirely meaningless. However, if it is meaningful, then I sense that it could be very important.

At the time I was largely unfamiliar with the details of the theory surrounding the distribution of prime numbers. I was familiar with the prime number theorem, but was only vaguely aware of the Riemann zeta function and the Riemann Hypothesis.

The central idea was essentially this:

In some previously unexplored context, the familiar 'shape' of the sequence of prime numbers is the result of a kind of dynamic or evolutionary process

and

the various mysterious analytic properties of the prime distribution arise from the nature of the dynamics involved in this process.

Obviously I realised that the distribution of primes exists, unchanging, outside historical time, but the idea had such an effect on me that I decided to keep an open mind, and to attempt to give it some kind of mathematical meaning.

In order to do this, I began exploring recent work concerning the distribution of primes, and discovered the following intriguing articles:

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

[2] B. Julia, "Statistical theory of numbers", from Number Theory and Physics (Springer Proceedings in Physics Volume 47, Luck et.al. eds., Springer, 1990)

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

To summarise very briefly:

Wolf discovered the presence of 1/f noise experimentally in the primes. He notes the work of Bak, et.al. which links 1/f noise to self-organised criticality. The paper ends with the astonishing question "Are the primes in a self-organized critical state?".

Julia uses thermodynamic thinking to construct an abstract numerical "gas" from the primes, in a very simple and natural way, such that its partition function is nothing but the Riemann zeta function. Note that the partition function is the fundamental object of study in statistical mechanics, much as the zeta function is the fundamental object of study in the analysis of the prime distribution.

Berry and Keating discuss work which demonstrates remarkable similarities between (i) the set of nontrivial zeros of the Riemann zeta function and (ii) the spectrum of energy levels (eigenvalues) of a quantum mechanical system whose underlying classical dynamics are chaotic and not time-reversible.

I have since discovered and studied many other articles related to these three, or in some way linking primes (often via the zeta function) with physics. In order to bring all of this fascinating research into one place I've created a website with numerous links, archived articles, abstracts, etc.

number theory and physics archive

I hope this may be of some interest or assistance to some of you. It is entirely independent from my own novel ideas on the subject.

Most significantly, [3] explained that the Riemann Hypothesis would be proven if a specific kind of QM system could be identified, whose spectrum exactly matched the 'heights' of the complex zeros of the zeta function. In other words, the distribution of primes, via the set of complex zeta zeros which encodes it, 'points to' or 'implies' one very particular dynamical system, and the task of proving the RH has been reduced to finding that system.

I couldn't help thinking that if something like an 'evolutionary dynamics' underlying the prime distribution could be meaningfully formulated, it could very possibly related to this mysterious dynamical system which Berry, et.al. seek. For if the Riemann dynamics hypothesised in [3] does exist in some sense, to what would it refer? What could it possibly be describing?

It would be deeply connected to the very nature of number, and would be categorically different from anything else ever formulated.

I have assembled a collection of (interesting, but mostly non-rigorous) notes describing in detail various aspects of my 'evolutionary' speculations, and possible clues regarding the nature of the dynamics involved.

Considerable time has been spent trying to make this as clear and readable as possible despite the unorthodox nature of the material. However, it is possible that none of the details will survive rigorous examination.

In any case, in these notes I pose certain precise mathematical questions which you are invited to consider. I'm quite confident that an attempt (successful or not) to answer these questions might lead to something new or interesting (however small).

back to 'prime evolution' notes