## Random Matrices and the Riemann Hypothesis

I made it to my second post!  This is the greatest achievement of my blogging life so far.

Having never even heard of random matrix theory before last week, I have recently been hearing talk of it bandied about all over the place.  When I asked my supervisor why this might be, he replied that they were a “hot topic”, and went on to explain the recent connections that have been made between the distribution of zeros of the Riemann zeta function, and that of eigenvalues of large random matrices.  So I’m going to write a bit about it here.

Random matrices are intuitively exactly what you might first suspect they are: matrices with “random” entries.  Of course, it is technically meaningless to say that an element of a matrix is random (hence the inverted commas), but what we can do is to use probability theory to rigorously define an analogous concept.  Without going into in too much depth, we can think of a random matrix as being a matrix with elements that are random numbers from some probability distribution (that is, random variables).  So an $n\times n$ random matrix is actually a collection of all the possible $n\times n$ matrices, along with probability density functions telling us how likely each of these is to occur.

One of the reasons that the study of these is currently in vogue is their usefulness in modelling chaotic systems.   Traditionally, systems in classical or quantum mechanics were seen to be governed by some differential equation, giving a means to accurately predict the state of the system at some future point given only its present condition.  However, physicists have come to the realisation that this is a naive point of view, and that in reality “most” systems have some level of instability – or chaos – inherent in them, such that small perturbations in the initial conditions produces large differences in the future state.

It has been found that the sequence of states of such systems can be described by eigenvalues of random matrices.  One area in which this method has been proven particularly useful is nuclear physics, in which the dynamics of nuclei in a reaction become so complex that it becomes impossible to keep track of individual states, and it is meaningful to instead model average or probabilistic properies of the system.

This is all very vague, I know.  But it isn’t really what I wanted to talk about.  Changing tack somewhat, the Riemann zeta function is defined to be the analytical continuation of:

$\zeta (z)=\displaystyle\sum _{n=1} ^\infty n^{-z}=\displaystyle\prod_p (1-p^{-z})^{-1},$

where the product on the right is over every prime$p.$Arguably the most famous unsolved problem in mathematics is to find a proof of the Riemann Hypothesis, which conjectures that every zero of this function has real part $\frac{1}{2}$.  If this were true, it would imply that the prime numbers have a random but regular distribution; the central role that the mysterious primes play in pure mathematics – not to mention their importance for more practical pursuits such as cryptography and statistics – goes some way to explaining the problem’s importance.

Now, assuming the Riemann hypothesis to be true, let $z=\frac{1}{2}+b_n i$ be the non-trivial zeros of the zeta function.  It turns out that the $b_n$ are distributed on the real line according to the same laws as the eigenvalues of certain large random matrices.  To be precise: the two-point correlation function of the zeros $z$ of the Riemann zeta function on the line $\Re (z)=\frac{1}{2}$ is the same as that of the eigenvalues of a random Hermitian matrix taken from the Gaussian unitary ensemble (unitary matrices satisfying certain constraints).

This can’t possibly just be a coincidence, and so we can add nuclear physics to the extraordinarily large list of fields in which the Riemann zeta function crops up.   There are even more surprising ones than this though: music, for example.  And I don’t just mean in the anological sense of Marcus du Sautoy’s book “The Music of the Primes”; the function actually has direct applications to the study of tuning systems.

But that is very much another story…perhaps I will write a post on mathematics in music someday soon.