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<h1 align=center> Dynamical zeta-functions
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 For several decades now there has been interest
amongst physicists and mathematicians to study the statistics of
the eigenvalue spectra of physical systems - especially those with
a classical counterpart which displays chaotic behaviour.  One
particular goal is to try to detect in the statistical
distribution of a set of energy eigenvalues an indication of
whether the corresponding classical system behaves chaoticly or
integrably.  The answer was suggested by Berry and Tabor in 1977,
and examined in depth by Bohigas, Giannoni and Schmit
[
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<a href="http://www.ams.org/msnmain?fn=130&s1=85f:58034&pg1=MR">MR 85f:58034</a>
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(non-chaotic) systems have a spectrum of uncorrelated eigenvalues,
the spectra of classically chaotic systems show the characteristic
correlations seen in the spectra of ensembles of random matrices.
In particular, the semiclassical spectra of physical systems which
are not symmetric under time-reversal have the same local
statistics (those of the ensemble of unitary matrices, $U(N)$,
with Haar measure) as the zeros high on the critical line of the
Riemann zeta function and other $L$-functions.

Thus for a given physical system which possesses the correct
symmetries, we can construct a zeta function which will have zeros
correlated on a local scale like those of the Riemann zeta
function. For a system with a Hamiltonian operator $H$ and a set
of eigenvalues $E_n$, satisfying
$$H\psi_n(\bf{r})=E_n\psi_n(\bf{r}),$$ with wavefunctions
$\psi_n$, a natural function to study is the spectral determinant
$$\Delta(E) \equiv \det[A(E,H)(E-H)] = \prod_j[
A(E,E_j)(E-E_j)],$$ where $A$ has no real zeros and is included so
as to make the product converge.

A semiclassical expression for the spectral determinant can be
given using only classical attributes of the system in question.
Following Berry and Keating [
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<a href="http://www.ams.org/msnmain?fn=130&s1=92m:81053&pg1=MR">MR 92m:81053</a>
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$$\Delta(E)\sim B(E) \exp(-i\pi \overline{N}(E)) \prod_p \exp
\left( -\sum_{k=1}^{\infty} \frac{\exp(ikS_p/\hbar)}
{k\sqrt{|\det(M_p^k-1)|}} \right).$$ Here $B(E)$ is a real
function with no zeros and
$$\frac{d \overline{N}(E)}{dE}=\overline{d}(E),$$ the mean density
of eigenvalues.  The sum is over the periodic orbits of the
classical system, $S_p$ is the action of the orbit $p$, and $M_p$
is the monodromy matrix which describes flow linearized around the
orbit.

We can now define the dynamical zeta function $$ Z(E)=\prod_p \exp
\left( -\sum_{k=1}^{\infty} \frac{\exp(ikS_p/\hbar)}
{k\sqrt{|\det(M_p^k-1)|}} \right).$$  Since $B(E)$ has no zeros,
the zeros of $Z(E)$ are the eigenvalues of the system under
investigation.
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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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