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%\topmatter\title The Selberg zeta function from the geometric, dynamic and arithmetic points of view\endtitle
\topmatter\title New points of view on the Selberg zeta function\endtitle
\author Don Zagier\endauthor 
\abstract The classical Selberg zeta function for the modular group $\G_1=\SL(\Z)$ 
relates the set of traces of hyperbolic elements of $\G_1$ and the set of eigenvalues of Maass wave
forms for~$\G_1$.  In recent years new connections have emerged relating these two objects to two new
objects: the spectrum (and in particular the eigenvalues~$\pm1$) of a certain ``transfer operator"~$\L_s$
defined by D.~Mayer and the analytic solutions of the ``three-term functional equation" $\psi(x)
=\psi(x+1)\pm x^{-2s}\psi(x\i+1)$ introduced by J.~Lewis.  We will give a short survey of these
connections and also briefly discuss numerical aspects and extensions to other groups.\endabstract\endtopmatter


\section{\S1.  The Selberg trace formula and the Selberg zeta function}
If $\G$ is a discrete subgroup of cofinite volume of $G=\SL(\R)$, acting in the usual way on the
upper half-plane $\H$, then we have two interesting  associated sequences of positive real numbers.
The first is the ``length spectrum" $\;0<l_1\le l_2\le\cdots$, which can be defined for $\G$ torsion-free and
cocompact as the set of lengths of all closed geodesics in $\G\bs\H$ (counting multiplicities), and in 
the general case as the set of numbers $\,\log N(\g)\,$ for $\g$ ranging 
over all conjugacy classes of hyperbolic elements of~$\G$, where $N(\g)$ is defined as $\v^2$ for
$\g$ conjugate to $\pm\smat\v 0 0 {1/\v}$ with $\v>1$.  The second is the set $\;0=\l_0<\l_1\le\l_2\le\cdots\,$
of eigenvalues of the hyperbolic Laplace operator  $\D=-y^2(\p^2/\p x^2+\p^2/\p y^2)$  on $\G\bs\H$. (Here
$x$ and $y$ denote the real and imaginary part of the coordinate $z\in\H$ as usual.)  One has information
about the asymptotics of the set $\{\l_j\}$, but the eigenvalues themselves are in general quite mysterious.
For the full modular group $\G_1=\SL(\Z)$, for instance, many eigenvalues have been computed numerically (the first 
and third have the values $\l_1=91.14134\cdots$ and $\l_3= 190.13154\cdots$), but not a single eigenvalue is known 
``in closed form."  The length spectrum, on the other hand, is fairly easily computable and in the case of 
$\G=\G_1$ consists essentially of the logarithms of units of all real quadratic fields, with multiplicities 
equal to the class numbers of these fields.

The Selberg trace formula relates these two invariants of the action of $\G$. Roughly speaking, it has the form
$$ \sum_{j=0}^\infty H(\l_j)\= \sum_{j=1}^\infty \widetilde H(l_j)\,,\tag1$$
where $H$ is a sufficiently nice ``test function" (analytic and of suitable decay in a suitable
neighbourhood of $\R_+$, but otherwise arbitrary) and $\widetilde H$ an explicitly given integral
transform of $H$.  (Actually, the formula is a little more complicated: the term $\widetilde H(\l_j)$
corresponding to the conjugacy class of a hyperbolic element which is the $n$th power of a primitive
element must be divided by $n$, and one has to add to the right-hand side further contributions coming
from the parabolic elements and the elements of finite order of the group $\G$.)

By choosing for $H$ a suitable function depending on a complex parameter $s$ and exponentiating both sides
of~(1), one obtains the {\it Selberg zeta function}, $Z_\G(s)$.  The expression for $Z_\G(s)$ as a product
over hyperbolic elements takes the form
$$  Z_\G(s)\=\prod_\g\prod_{m=0}^\infty\bigl(1\,-\,N(\g)^{-s-m}\bigr)\qquad(\Re(s)\gg0),\tag2$$
where $\g$ in the first product ranges over all primitive hyperbolic elements of $\G$ (or equivalently, 
all primitive geodesics of $\G\bs\H$), while the expression for $Z_\G(s)$ as a product over eigenvalues 
of the Laplace operator shows that $Z_\G(s)$ has a meromorphic continuation in $s$ and has zeros (with the appropriate
multiplicities) at all the {\it spectral parameters} of $\G$, defined as the numbers $s_j$ with $s_j(1-s_j)=\l_j$.
(Notice that there are two $s_j$ for each eigenvalue $\l_j$, the values corresponding to the two eigenvalues
given above for $\G=\G_1$ being $\h\pm9.53369526\cdots i$ and $\h\pm13.779751\cdots i\,$.)  

In this note we will concentrate mostly on the case of the full modular group $\G_1=\SL(\Z)$ and its order~2
extension $\Gp=\GL(\Z)$, which can be made to operate on $\H$ by letting $\g=\smat abcd$ act by
$z\mapsto(a\bar z+b)/(c\bar z+d)$ if $\det(\g)=-1$.  The eigenvalues of the Laplace operator for $\Gp$ are
just the {\it even} eigenvalues for $\G_1$, i.e., those corresponding to eigenfunctions which are symmetric 
under $z\mapsto-\bar z$ (of the two eigenvalues for $\G_1$ given above, the first corresponds to an odd 
eigenfunction and the second to an even one), and the corresponding spectral parameters are zeros of the
Selberg zeta function $Z_{\Gp}(s)$, which is defined by a formula like (2) but with $\g$ now ranging over the
primitive hyperbolic elements of $\Gp$ and with the factor $1-N(\g)^{-s-m}$ replaced by $1-(-1)^mN(\g)^{-s-m}$
whenever $\g$ has determinant $-1$.

\section{\S2. The Mayer transfer operator $\L_s$}
A surprising new interpretation of the Selberg zeta function was found by D.~Mayer [9].  Let $\Di$ denote the
open disc of radius~3/2 and center~1 and $\V$ the space of functions which are holomorphic in $\Di$ and extend
continuously to its boundary.  Then Mayer defines an operator $\L_s:\V\to\V$ by the formula
$$ \L_s\phi(z)\=\sum_{n=1}^\infty\frac1{(z+n)^{2s}}\,\phi\bigl(\frac1{z+n}\bigr)\qquad(\phi\in\V,\;\,z\in\Di)\,.\tag3$$
The series converges for $\Re(s)>\h$ for any $\phi\in\V$, and can\ be continued meromorphically to all $s$.
(Write $\phi(z)$ as the sum of a polynomial of degree $k$ and a function which is $\,\text O(z^{k+1})\,$ as $z\to0$;
then the RHS of (3) is the sum of a finite linear combination of Hurwitz zeta functions, whose meromorphic
continuation is well known, and an infinite sum which converges in the half-plane $\Re(s)>-k/2$.)  Mayer shows
that this operator has eigenvalues converging rapidly to zero (``nuclear"); in particular, $\L_s$ is of trace
class and $1-\L_s$ has a determinant in the Fredholm sense, and similarly for~$\L_s^2$. Mayer's result is then:
\proclaim{Theorem} We have the identities
$$ Z_{\G_1}(s)\=\det(1-\L_s^2),\qquad Z_{\Gp}(s)\=\det(1-\L_s)\,.\tag4$$ \endproclaim
\noindent (Actually, Mayer gave the first of these statements, for the usual modular group. The second statement,
corresponding to the even eigenfunctions, was proved by Efrat~[2].)  

Mayer's proof came from the theory of dynamical systems, and more specifically from the dynamics of the Gauss map 
(continued fraction map) $F:[0,1)\to[0,1)$ which assigns to any $x>0$ the fractional part of $1/x$.  For each 
complex number $s$ with $\Re(s)>\h$ and each positive integer $n$ we then have the {\it partition function}
$$ Z_n(F,h_s)\=\sum_{x\in[0,1),\;F^nx=x} h_s(x)h_s(Fx)\cdots h_s(F^{n-1}x)\,,$$
where $h_s$ is the weighting function $h_s(x)=x^{2s}$ and the sum is over all periodic points of $F$ of order
dividing $n$.  The relationship with Mayer's operator $\L_s$ (``transfer operator") comes from the fact that the
inverse images of $x\in[0,1)$ are the points $(n+x)^{-1}$ ($n=1,\,2,\ldots$), and using this one can show that
$$ Z_n(F,h_s)\=\Tr(\L_s^n)\,-\,(-1)^n\,\Tr(\L_{s+1}^n)\,.$$
On the other hand, the periodic points of $F$ correspond to the periodic continued fractions of real quadratic
irrationalities and hence to closed geodesics in $\H/{\Gp}$, and using this correspondence one finds that 
the sub-product of (2) corresponding to $m=0$ for the group $\G=\G_1$ is equal to $\,\exp(\sum_{n=1}^\infty Z_{2n}(F,h_s)/n)$.
(Here only even indices occur because the map $x\mapsto(n+x)^{-1}$ has determinant $-1$. For $\Gp$ all indices occur.)
Putting these facts together, one obtains (4) after a short calculation.  A somewhat more elementary version of
the proof, based on the reduction theory of elements of $\Gp$, is presented in~[4].


\section{\S3. Period theory and the Lewis correspondence}

The final ingredient of our story is the discovery made some years ago by John Lewis [3] that one can
associate to an eigenfunction of the Laplace operator for $\Gp$ (i.e., an even eigenfunction
for $\G_1$) with spectral parameter $s$ a holomorphic solution in $\C\sm(-\infty,0]$ of
the {\it three-term functional equation}
 $$ \psi(z)\=\psi(z+1)\,+\,z^{-2s}\,\psi(1+1/z)\,.\tag5$$
This result was extended in several ways in subsequent joint work [5]; in particular, it was shown 
that the correspondence is reversible, that it is enough to consider real-analytic solutions of~(5) 
on the positive real axis, and that there is a similar result for odd eigenfunctions of the Laplacian. 
The final result can be stated as follows.  
\proclaim{Theorem} Let $s$ be a complex number with $\Re(s)=\h$.  Then there is a canonical
isomorphism between the space of even or odd eigenfunctions of the Laplace operator for $\G_1$ with 
eigenvalue $s(1-s)$ and the space of real-analytic functions $\psi$ on $(0,\infty)$ satisfying
the even or odd three-term functional equation
 $$ \psi(x)\=\psi(x+1)\,\pm\,x^{-2s}\,\psi(1+1/x)\tag6$$
and the growth condition $\lim\limits_{x\to\infty}\psi(x)=0$.  \endproclaim
\noindent Note that the theorem implies in particular
that the space of real-analytic solutions of (6) satisfying the growth condition $\psi(x)=\text o(1)$
as $x\to\infty$ is finite-dimensional for all $s$, and non-zero only for countably many $s$.  This is
especially surprising since, as is shown in the paper, the space of real-analytic solutions of~(6)
satisfying the apparently only slightly weaker condition $\psi(x)=\text O(1)$ as $x\to\infty$ is
infinite-dimensional for every $s$.  We also mention that there is also a uniform version of the bijection
of the theorem, including both the even and odd cases, in which the two functional equations (6) are 
replaced by the ``unified" functional equation $\psi(x)\=\psi(x+1)+(x+1)^{-2s}\,\psi(x/(1+x))$.  

The above theorem is the exact analogue of the classical theory of Eichler, Shimura and Manin of period
polynomials of modular forms, according to which one can canonically associate to a classical cusp form
$f(z)$ of weight $2k$ on $\G_1$ two polynomials, one even and one odd, which satisfy the above even
and odd three-term functional equations (in the opposite order) with $s=1-k$, obtaining in this
way an isomorphism between the space of cusp forms and the spaces of polynomial solutions of each
of the two functional equations.  (More precisely, for the even polynomials one obtains a codimension
one subspace of the space of polynomial solutions of the odd functional equation, the full space
being obtained only if one includes an Eisenstein series.)  The correspondence is obtained
by associating to $f(z)$ its {\it Eichler integral} $f^*(z)$, defined most easily by the formula 
$f^*(z)=\sum_{n=1}^\infty n^{1-k}a_n\,e^{2\pi inz},$ where $f(z)=\sum_{n=1}^\infty a_ne^{2\pi inz}$ 
is the Fourier expansion of $f$; then the function $\psi(z)=f^*(z)-z^{2k-2}f^*(-1/z)$ turns out to
be a polynomial satisfying the above-mentioned ``unified" three-term functional equation with $s=1-k$, 
and its even and odd parts are the desired polynomials satisfying the two equations (6).  

The construction here proceeds similarly, although the formulas are somewhat more complicated and the proof 
is considerably longer.  If $f(z)$ is an eigenfunction of the Laplace operator for $\G_1$ (a so-called {\it Maass 
wave form}) with spectral parameter~$s$, then, as is well known, $f$ has a Fourier expansion of the form
$$ f(z)\=\sqrt y\;\sum_{n=1}^\infty A_n\,K_{s-\h}(2\pi ny)\,\cos(2\pi nx)\qquad(z=x+iy\in\H) \tag7$$
in the even case and a similar expansion with ``sin" instead of ``cos" in the odd case, where $K_\nu(t)$
is the usual $K$-Bessel function and the coefficients $A_n$ are complex numbers which grow at most
polynomially in $n$.  Although this function is non-holomorphic, we define its ``Eichler integral"
to be the {\it holomorphic} function $f^*$ in the upper half-plane given by
$$f^*(z)\=\sum_{n=1}^\infty n^{s-\h}\,A_n\,e^{2\pi inz}\qquad(z\in\H)\,.\tag8$$
We extend this function to $\C\sm\R$ by setting $f^*(-z)=-f^*(z)$ in the even case (resp.~$f^*(-z)
=f^*(z)$ in the odd case).  Then by relating the Mellin transforms of the two functions $f^*(\pm iy)$ ($y\in\R_+$) 
to the $L$-series of $f$ and using the functional equation of this $L$-series, one shows that the function
$\psi(z)$ defined in $\C\sm\R$ by the formula $\psi(z)=f^*(z)-z^{-2s}f^*(-1/z)$ (i.e., by
exactly the same formula as in the classical case, with $k$ replaced by $1-s$) extends holomorphically
from $\C\sm\R$ to the cut plane $\C\sm(-\infty,0]$. The function $\psi$ of the theorem is then simply the
restriction of this extended function to the positive real axis.  That $\psi$ satisfies the three-term functional
equation in $\C\sm\R$ is a trivial consequence of its definition and the periodicity of $f$, and would be true
for any function in $\H$ with a Fourier expansion of the form~(7), i.e., for any eigenfunction of the Laplace
operator which is invariant merely under $z\mapsto z+1$ rather than under the whole modular group. The 
non-trivial property, which holds only if the function defined by (7) is invariant under the full group
$\G_1$, is the fact that $\psi$ extends analytically across the positive real axis. 

For the converse direction, if $\psi$
is a real-analytic solution of (6) satisfying the growth condition in the theorem, one first shows by a
``bootstrapping" argument (successive extension of $\psi$ to larger and larger domains) that the function
$\psi$ extends holomorphically from $\R_+$ to all of $\C\sm\R$ and satisfies a suitable growth condition
near the cut; then the functional equation of $\psi$ shows that the function $f^*(z)$ defined by the formula
$f^*(z)=\psi(z)+z^{-2s}\psi(-1/z)$ is periodic with period~1, and hence has a Fourier expansion of the form~(8)
for some coefficients $A_n\in\C$, and one then shows that the function $f(z)$ defined by the Fourier expansion~(7)
with the same coefficients $A_n$ is indeed $\G_1$-invariant.  The key fact here, which has no analogue in the 
classical case, is the simple algebraic identity that the two transformations $\psi(z)=f^*(z)-z^{-2s}f^*(-1/z)$
and $f^*(z)=\psi(z)+z^{-2s}\psi(-1/z)$ are inverses of one another up to a non-zero scalar factor for
any $s\in\C\sm\Z$.


\section{\S4. Interrelationships and generalizations}
We now consider the relationship between the various objects we have described.  In \S1 we saw how the length spectrum
of $\G\bs\H$ can be encoded in an infinite product, the Selberg zeta function, whose meromorphic continuation
has zeros at the spectral parameters $s_j$ of $\G$.  In \S2 we saw how this zeta function, for $\G=\Gp$ or
$\G_1$, could be written as the Fredholm determinant of $1-\L_s$ or $1-\L_s^2$, respectively.  But this Fredholm 
determinant is given by a convergent product $\prod_\nu(1-\beta_\nu(s))$ or $\prod_\nu(1-\beta_\nu(s)^2)$, where
$\beta_\nu(s)$ are the eigenvalues of the operator $\L_s$, so its vanishing is equivalent to the statement that
$\L_s$ has the eigenvalue 1 or $\pm1$.  We thus obtain as a direct corollary of the theorem in \S2 the following
statement: {\it If $s$ is the spectral parameter for an even (resp.~odd) eigenfunction of the Laplace operator 
for $\SL(\Z)$, then there exists a non-zero function $\phi\in\V$ satisfying $\L_s\phi=\phi$ (resp.~$\L_s\phi=-\phi$).}

The relationship between this statement and the period theory of Maass wave forms which we described in the
last section is now very simple:  It is obvious from the definition~(3) that the operator $\L_s$ satisfies 
the functional equation $\L_s\phi(x)=(x+1)^{-2s}\phi(1/(x+1))+\L_s\phi(x+1)$, so if $\phi\in\V$
is an eigenfunction of $\L_s$ with eigenfunction $\pm1$ then we have the identity 
$\phi(x)=\phi(x+1)\pm(x+1)^{-2s}\phi(1/(x+1))$, and after the simple change of variables $\phi(x)=\psi(x+1)$
this is precisely equation~(6).  

We can therefore summarize the whole story by saying that there are very close relationships between
{\it four} a priori very different objects, as indicated in the following diagram,
$$\xymatrix{*++{\boxed{{\text{A. Set of lengths of}}\atop{\text{geodesics on }\G_1\bs\H}}}
  \ar[rr]\ar[d]&&*++{\boxed{{\text{B. Spectrum of the}}\atop{\text{Laplace operator }\D}}}\ar[d]\ar[ll]\\
  *++{\boxed{{\text{C. Spectrum of the}}\atop{\text{Mayer operator }\L_s}}}\ar[rr]\ar[u]
  &&*++{\boxed{\text{D. Lewis's three-term}\atop{\text{functional equation}}}}\ar[u]\ar[ll]}$$
with (A) and (B) being related by the Selberg trace formula, (A) and (C) by the theory of transfer operators
applied to the dynamical system associated to continued fractions, (B) and (D) by a generalization to the case 
of Maass wave forms of the classical theory of periods of modular forms, and (C) and (D) by the fact that an 
eigenfunction of $\L_s$ with eigenvalue $\pm1$ is, up to a shift by 1, a solution of the functional equation~(6).

It is also natural to ask what happens when $\G_1$ is replaced by some other discrete group $\G\subset\SL(\R)$.
The relationship between the lengths of geodesics and the eigenvalues of the Laplacian for $\G\bs\H$ given by 
the Selberg trace formula works equally well, as discussed in \S1, but  both the Mayer operator and the 
three-term functional equation depend on the special properties of the group $\G_1$, and specifically on the 
fact that it is generated by the two transformations $S:z\mapsto-1/z$ and $T:z\mapsto z+1$.
If $\G$ is a subgroup of $\G_1$ of finite index $N$, then we can reduce to the case of $\G_1$ by replacing the
functions on which $\L_s$ acts and the function $\psi$ by vector-valued functions of length $N$, with
coordinates indexed  by the set of cosets $\G\bs\G_1$, and with the appropriate modifications of the equations 
(for instance, in the functional equation~(6) the coordinates of the two terms on the right must be permuted
according to the actions of the two matrices $\smat1101$ and $\smat1011$ on $\G\bs\G_1$), and everything goes
through.  This is the point of view adopted in the thesis of Martin~[8], in which a theory analogous to the one
in~[5] is developed for the case of holomorphic modular forms of weight~1, a case not covered by the classical 
period theory of Eichler-Shimura-Manin, and also in the recent work of Manin and Marcolli~[7].  However, there 
is also a more intrinsic approach which works also if the group $\G$ is not contained in, or even 
commensurable with, the modular group $\G_1$.  In the case of the Mayer
operator, this requires considering the analogue of the continued fraction expansion appropriate to the
group $\G$, based on the geometry of a chosen fundamental domain for $\G$, and replacing $\L_s$ by the transfer
operator appropriate to this new continued fraction.  For example, if we observe that $\L_s$ is really attached
to $\Gp$ rather than to $\G_1$ (as seen by the fact that the Selberg zeta function for $\G_1$ corresponds to
the determinant of $1-\L_s^2$ rather than $1-\L_s$), then we should already modify $\L_s$ even for the case
of the subgroup $\G_1$; the right modification of~(3) is the operator 
$\phi(z)\mapsto\sum_{n=2}^\infty(n-z)^{-2s}\phi(1/(n-z))$, whose fixed points correspond via
$\phi(z)=(1-z)^{-2s}\psi(1/(1-z))$ to the solutions of the ``unified" functional equation mentioned above. For 
the Lewis correspondence, the most convenient language in which to describe the generalization to arbitrary~$\G$
is that of cohomology of groups.  We indicate here very briefly how this works. (The details will be given in~[6].)

Define $V_s^\infty$ to be the space of $C^\infty$ functions $\psi:\R\to\C$ such that the function $|x|^{-2s}\psi(1/x)$ 
on $\R\sm\{0\}$ extends to a $C^\infty$ function on $\R$ (or equivalently, that the function 
$\Psi(x,y):=|y|^{-2s}\psi(x/y)$ on $\R^2\sm\{(0,0)\}$ is $C^\infty$), and let $V_s^\o$ be the subspace 
defined the same way but with ``$C^\infty$" replaced by ``real-analytic."  We define an intermediate space $\Vh$ 
as the space of functions in $V_s^\infty$ which are real-analytic except at a finite subset of $\Bbb P^1(\R)$.  
The group $G=\SL(\R)$ acts on all three spaces by $(\psi|g)(x)=|cx+d|^{-2s}\psi(g(x))$ for $g=\smat abcd\in G$ 
(or equivalently, by $\Psi\mapsto\Psi\circ g$).  In~[5] it is shown that the function $\psi$ occurring in the
theorem in~\S2, if extended from $\R_+$ to $\R\sm\{(0,0)\}$ by $\psi(x)=\mp\psi(-x)$ (or equivalently by
$\psi(x)=-|x|^{-2s}\psi(-1/x)$), belongs to $V_s^\infty$ and hence also to $V_s^{\o/2}$, and now the three-term
functional equation implies exactly that this extended function satisfies the equations $\psi+\psi|S=
\psi+\psi|U+\psi|U^2=0$, where $U=ST$ is the standard element of $\G_1$ of order~3 which together with the
involution $S$ generates this group.  These equations say that the map $S\mapsto\psi$, $T\mapsto0$ extends
to a cocycle on $\G$ with values in $\Vh$, and the fact that $T$ maps to~0 says that this is a parabolic
cocycle (one which sends every parabolic element $\g\in\G$ to an element in the image of $1-\g$). The theorem
of \S2 can then be reformulated as the statement that the parabolic cohomology group $H^1_{\text{par}}(\G_1,\Vh)$
is isomorphic to the space of Maass wave forms with spectral parameter $s$ for all complex numbers $s$ with
real part 1/2, and the generalization to other $\G$ is simply the same assertion for $H^1_{\text{par}}(\G,\Vh)$
and the spectral parameters for $\G$.  (If $\G$ is contained in $\G_1$ and one uses the point of view of
vector-valued functions mentioned above, then one would look instead at $H^1_{\text{par}}(\G_1,(\Vh)^{\G\bs\G_1})$;
this is the form in which the corresponding theorem for holomorphic forms of weight~1 occurs in [8].) Finally,
it turns out that one can change the above cocycle by a coboundary to obtain a new cocycle, no longer sending~$T$
to~0, with values in the smaller space $V_s^\o$; these new cocycles are no longer parabolic, but satisfy the 
weaker ``semi-parabolic" condition that their value at every parabolic element $\g$ of $\G$ belongs to 
$\Vh|(1-\g)$, and the cohomology group $H^1_{\text{par/2}}(\G,V_s^\o)$ defined by such semi-parabolic
real-analytic cocycles modulo coboundaries is again isomorphic to the
space of Maass wave forms on $\G$ with spectral parameter~$s$.

% \newpage

\section{\S5. Computational aspects}
We end this survey by discussing to what extent the new points of view on the Selberg zeta function
are usable from a practical or computational point of view.  The usual Selberg trace formula, both
in its general form and in the special case defining the Selberg zeta function, has the drawback that
it is very hard to use computationally: both the product~(2) and the Weierstrass product expressing
$Z_\G(s)$ in terms of its zeros and poles are either divergent or only slowly convergent, and it is
not easy to calculate actual values of the Selberg zeta function or to use the Selberg trace formula 
to get explicit information about the eigenvalues of the Laplace operator from the lengths of
geodesics or vice versa.  On the other hand, as we mentioned in \S4, the eigenvalues $\beta_\nu(s)$
of the operator $\L_s$ tend to zero rapidly, so that the infinite products  $\prod(1-\beta_\nu(s))$
and $\prod(1-\beta_\nu(s)^2)$ defining the Fredholm determinants $\,\det(1-\L_s)=Z_{\Gp}(s)$ and
$\,\det(1-\L_s^2)=Z_{\G_1}(s)$ are rapidly convergent.  Hence if we have a good algorithm to compute the
eigenvalues $\beta_\nu(s)$ then we also have a good method to compute both the values and the zeros of 
these Selberg zeta functions.

In fact such an algorithm exists.  It was developed by Babenko (cf.~[1]) to study the case $s=1$,
corresponding to a famous problem of Gauss (in a letter to Laplace, 1812) and Kuzmin (1928) concerning the 
distribution of the ``tails" of the continued fraction expansions of real numbers, and the method 
extends in a fairly straightforward way to other values of $s$.  On the one hand, one can obtain a good
representation of the operator $\L_s$ by showing that it is conjugate to (and hence has the same
eigenvalues as) the integral operator on an appropriate space of functions on $(0,\infty)$ given by the kernel function
$$K_s(x,y)\=\frac{J_{2s-1}\bigl(2\sqrt{xy}\bigr)}{\sqrt{(e^x-1)(e^y-1)}}\,,$$
where $J_\nu(x)$ is a $J$-Bessel function.  (This is shown in Babenko's paper for $s=1$ and a modification 
of the same proof works in general. It also occurs, in a somewhat different formulation, in Lewis's paper~[3].)  
This representation as an integral operator can be used to establish the compactness and various other properties
of the operator $\L_s$.  On the other hand, the operator $\L_s$ sends the function $A_k(x)=x^{k-1}$ ($k=1,\,2,\,\dots$)
to $B_k(x)=\zeta(x+1,k+2s-1)$, where $\zeta(\a,\nu)=\sum_{n=1}^\infty(n+\a)^{-\nu}$ denotes the Hurwitz zeta
function.  If one takes $m$ well-chosen points $x_1,\ldots,x_m$ in $[0,1]$ (a good choice is the set of
points $x_j=\sin^2((2j-1)\pi/4m)$), then it turns out that the eigenvalues of the $m\times m$ matrix $A^{-1}B$, 
where $A$ is the Vandermonde matrix $\bigl(A_k(x_j)\bigr)_{1\le j,k\le m}$ and $B$
the matrix $\bigl(B_k(x_j)\bigr)_{1\le j,k\le m}$, 
are good approximations to the desired eigenvalues $\b_\nu(s)$ of $\L_s$.  (More precisely, the matrix $AB^{-1}$
has some eigenvalues $\b_{\nu,m}(s)$ which tend exponentially rapidly to $\b_\nu(s)$ as $m\to\infty$, as well as
some ``spurious" eigenvalues which tend slowly, like a negative power of $m$, towards zero as $m$ goes to infinity.)  
As an example, we list in Table~1 the approximate values of the first 19 eigenvalues $\b_\nu=\b_\nu(1)$
of $\L_1$ obtained in this way using values up to $m=50$.  (The first five of these, based on a calculation 
with $m=16$ and correct to about 8 digits, were already given in~[1], p.~367, but the table given there also 
contains one of the spurious eigenvalues which should be discarded.)  As a check on the accuracy of the
computation, one can compare the sum of the numbers $\b_\nu(s)$ and the sum of their squares with the values 
of $\,\Tr(\L_s)\,$ and $\,\Tr(\L_s^2)$, both of which can be computed exactly. 

\newpage
$$\align \b_1 &\= \phantom{-}1.0000000000000000000000000000,\\
         \b_2 &\= -0.3036630028987326585974481219 \\
         \b_3 &\= \phantom{-}0.1008845092931040753056376425, \\
         \b_4 &\= -0.03549615902165984540889163041, \\
         \b_5 &\=  \phantom{-}0.01284379036244026481516090344, \\
         \b_6 &\= -0.004717777511571031073863594144, \\
         \b_7 &\= \phantom{-}0.001748675124305511914510617525, \\
         \b_8 &\= -0.0006520208583205029031816912077, \\
         \b_9 &\=   \phantom{-}0.0002441314655245138906981706362,  \\
         \b_{10} &\=  -0.00009168908376859566896987721814 \\
         \b_{11} &\= \phantom{-}0.00003451654616347253214653427704, \\
         \b_{12} &\=  -0.00001301769787581775595712876501, \\
         \b_{13} &\=  \phantom{-}0.000004916782323649260641924415470, \\
         \b_{14} &\=  -0.000001859377799671070019642381811, \\
         \b_{15} &\=  \phantom{-}0.0000007038108595990962920678071171, \\
         \b_{16} &\=  -0.0000002666477929249816513412730701, \\
         \b_{17} &\=  \phantom{-}0.00000005079864542122092292865860954,\\
         \b_{18} &\=  -0.00000003831081550679296190236314295,\\
         \b_{19} &\=  \phantom{-}0.00000001400621008979342385785713695. \\
         \\ &\qquad\qquad\text{Table 1. Eigenvalues of $\L_1$} \endalign $$

Applying this method with $s$ of the form $\h+it$ with $t$ real, we can compute the eigenvalues
$\b_\nu(s)$ to high accuracy for small values of $t$.  If we plot the curve of the largest (in
absolute value) eigenvalue $\b_1(\h+it)$ in the complex plane, then we indeed find that within 
the accuracy of the computation it passes through the points $-1$ and $+1$ for 
$t\approx9.53369526135355746$ and $t\approx13.77975135189074$, respectively, in accordance with 
Mayer's theorem from~\S2 and the known numerical values of the first odd and even spectral 
parameters for $\G_1$.  An interesting point here would be to find a direct argument showing
that the curve really passes exactly through the points $-1$ and $+1$, not just very near them.
It ought to be possible to do this using the functional equation of the Selberg zeta function,
as one does for the analogous statement for the zeros of the Riemann zeta function, but I have not
seen how to do this.

Finally, we mention that the numerical calculation described above could be used, for instance, to 
calculate to high accuracy the value of the logarithmic derivative at $s=1$ of the Selberg zeta function 
for $\G_1$, an invariant which plays a role in the work of Jorgenson and Kramer discussed at this conference.


\Refs

\ref \key 1 \by K.I. Babenko and S.P. Yurev \paper On a problem of Gauss
\pages 331--378 \jour  Sel. Math. Sov. \vol 2  \yr  1982   \endref

\ref \key 2 \by I. Efrat
\paper Dynamics of the continued fraction map and the spectral theory of $SL(2,\bold Z)$
\jour Invent. Math. \vol 114 \yr 1993 \pages 207--218 \endref 

\ref \key 3 \by J.B. Lewis
\paper Spaces of holomorphic functions equivalent to the even Maass cusp forms
\jour Invent. Math.\vol 127\yr 1997 \pages 271--306 \endref

\ref \key 4\by J.B. Lewis and D. Zagier
\paper  Period functions and the Selberg zeta function for the modular group \pages 83--97
\inbook The Mathematical Beauty of Physics 
\publ Adv. Series in Math. Physics {\bf24}, World Scientific\publaddr Singapore\yr 1997 \endref

\ref \key 5\by J.B. Lewis and D. Zagier \paper Period functions for Maass wave forms.~I
\jour Annals of Mathematics \vol 153 \yr 2001  \pages 191--258 \endref

\ref \key 6\by J.B. Lewis and D. Zagier \paper Period functions for Maass wave forms.~II
\jour \ in preparation \endref

\ref \key 7\by Yu.I. Manin and M. Marcolli \paper Continued fractions, modular symbols, and 
non-commutative geometry   \jour  Selecta Math., to appear  \endref

\ref \key 8\by F. Martin
\paper P\'eriodes de formes modulaires de poids~1
\jour Th\`ese, Universit\'e de Paris 7 \yr 2001 \endref

\ref \key 9\by D. Mayer
\paper The thermodynamic formalism approach to Selberg's zeta function for $PSL(2,\bold Z)$
\jour Bull. AMS \vol 25 \yr 1991 \pages 55--60 \endref

\endRefs
\end

