The Gaussian Unitary Ensemble Hypothesis
Pólya and Hilbert conjectured (with no evidence) that the nontrivial zeros of
the zeta function are of the form 1/2 + ib_{j}, where
the numbers b_{j} are the
eigenvalues of a naturally occurring Hermitian operator, hence real.
Hilbert had coined the term "spectrum" for the set of eigenvalues of
a Hermitian operator by analogy with the mysterious set of lines, called
spectra, which are the radiation frequencies of atoms. This was a fortuitous
terminology, because later, with the development of quantum mechanics, it
turned out that atomic spectra were actually spectra in the sense of operator
theory  the Hermitian operators in question being the Hamiltonians of the
dynamical systems representing the atoms in question.
The spectra of nuclei are more complicated than the spectra of atoms. A
nucleus absorbs a neutron and somewhat later undergoes radioactive decay.
The energies of such events are again quantized, but of such complexity
that only a statistical understanding of their nature is attempted. This
statistical line of inquiry was instituted by Wigner, pursued by Gaudin, Mehta
and Dyson, and is the subject of a book, Random Matrices, by Mehta.
There are three important theories of random hermitian matrices. These
model the Hamiltonians of random dynamical systems. If the dynamical
system is symmetric under time reversal, then the relevant theory is
that of the Gaussian Orthogonal Ensemble (GOE) or the
Gaussian Symplectic Ensemble (GSE), depending on whether the
total spin is an integer or halfinteger. On the other hand, the
Gaussian Unitary Ensemble (GUE) models random Hamiltonians
without time reversal symmetry. Since this is the statistical theory which is
relevant to the Riemann zeta function, this is the one we will describe.
The Gaussian Unitary Ensemble of degree N consists of the set of all
NxN Hermitian matrices, together with a certain probability measure.
This measure is the unique probability measure on the set of NxN Hermitian
matrices which is invariant under conjugation by unitary matrices, such that
the individual matrix entries are independent random variables.
Note that the first hypothesis is significant, but the second is artificial.
There is nothing sacred about these ensembles. Dyson has shown that the
essential features of the analysis can be carried out in a more natural and
general setting.
We now describe the GUE in specific terms. If i=j,
the ijth entry of the
matrix g is real, while if i<j, it is complex.
If i<j, let
g_{ij} = a_{ij} + ib_{ij}
The measure in question is Lebesgue measure times
.
The eigenvalues of random Hermitian matrices are not randomly distributed
in relation to each other. We recall the basic facts about the
Poisson distribution. Suppose that a series of points are distributed
at random along the line, with average density 1, and that the coordinates
of the points are independent random variables. Then the probability of
finding m points in a given interval of length x is
.
On the other hand, the eigenvalues of a random Hermitian matrix do not behave
like independent random variables, so the probability of finding more than
one eigenvalue in a short interval is much less than the Poisson distribution
would suggest. In short, the eigenvalues repel. This is shown by the
pair correlation function. Suppose that we examine an interval on which
the relative spacing of the eigenvalues has been adjusted so that the density
of zeros on this interval is one. If t is an eigenvalue in the
interval, the pair correlation function gives the probability density* of
finding a second eigenvalue near t+x. This function is:
Around 1970, Montgomery investigated the relative spacing of zeros of the zeta
function. He gave evidence that the pair correlation for the zeros of the zeta
function is exactly this function. He showed this to Dyson, who recognized
this as the pair correlation function for random Hermitian matrices in the
Gaussian Unitary Ensemble. Odlyzko gave impressive numerical evidence for
the GUE hypothesis, and in a tour de force
Rudnick
and Sarnak showed
that the "explicit formulas" that the zeros obey imply the GUE statistics;
assuming the Ramanujan conjecture, they prove this distribution for
zeros of Langlands Lfunctions. Thus on a small scale, the zeros of
the zeta function are distributed like the zeros of a random Hermitian
operator. On the large scale, there will be deviations from the GUE
distribution related to particular prime numbers.
Gutzwiller gave a trace formula in the setting of quantum chaos which
relates the classical and quantum mechanical pictures. Given a chaotic
(classical) dynamical system, there will exist a dense set of periodic orbits,
and one side of the trace formula will be a sum over the lengths of these
orbits. On the other side will be a sum over the eigenvalues of the
Hamiltonian in the quantummechanical analog of the given classical dynamical
system.
This setup resembles the explicit formulas of prime number theory.
In this analogy, the lengths of the prime periodic orbits play the role of the
rational primes, while the eigenvalues of the Hamiltonian play the role of
the zeros of the zeta function. Based on this analogy and pearls mined from
Odlyzko's numerical evidence, Sir Michael Berry proposes that there exists a
classical dynamical system, asymmetrical with respect to time reversal, the
lengths of whose periodic orbits correspond to the rational primes, and whose
quantummechanical analog has a Hamiltonian with zeros equal to the imaginary
parts of the nontrivial zeros of the zeta function. The search for such a
dynamical system is one approach to proving the Riemann hypothesis.
* A discussion of this (perhaps confusing) point between Mike Hardy and Dan Bump can be
found here.
[this is a slight adaptation of D. Bump's page at
http://match.Stanford.EDU/rh/gue.html]
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