NEW POINTS OF VIEW ON THE SELBERG ZETA FUNCTION

Don Zagier

Abstract. The classical Selberg zeta function for the modular group Γ_{1} = SL_{2}(ℤ) relates the set of traces of hyperbolic elements of Γ_{1} and the set of eigenvalues of Maass wave forms for Γ_{1}. In recent years new connections have emerged relating these two objects to two new objects: the spectrum (and in particular the eigenvalues ±1) of a certain “transfer operator” 𝔏_{s} defined by D. Mayer and the analytic solutions of the “three-term functional equation” ψ(x) = ψ(x + 1) ± x^{−2s}ψ(x^{−1} + 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.

§1. The Selberg trace formula and the Selberg zeta function

If Γ is a discrete subgroup of cofinite volume of G = SL_{2}(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} ≤ l_{2} ≤ ⋯, which can be defined for Γ torsion-free and cocompact as the set of lengths of all closed geodesics in Γ\H (counting multiplicities), and in the general case as the set of numbers log N(γ) for γ ranging over all conjugacy classes of hyperbolic elements of Γ, where N(γ) is defined as ε^{2} for γ conjugate to ±( \begin{matrix}ε & 0 \\ 0 & 1/ε \end{matrix} ) with ε > 1. The second is the set 0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ ⋯ of eigenvalues of the hyperbolic Laplace operator Δ = −y^{2}(∂^{2}/∂x^{2} + ∂^{2}/∂y^{2}) on Γ\H. (Here x and y denote the real and imaginary part of the coordinate z ∈ H as usual.) One has information about the asymptotics of the set {λ_{j}}, but the eigenvalues themselves are in general quite mysterious. For the full modular group Γ_{1} = SL_{2}(ℤ), for instance, many eigenvalues have been computed numerically (the first and third have the values λ_{1} = 91.14134 ⋯ and λ_{3} = 190.13154 ⋯), 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 Γ = Γ_{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 Γ. Roughly speaking, it has the form

∑_{j=0}^{∞} H(λ_{j}) = ∑_{j=1}^{∞} H̃(l_{j}) ,  (1)

where H is a sufficiently nice “test function” (analytic and of suitable decay in a suitable neighbourhood of ℝ_{+}, but otherwise arbitrary) and H̃ an explicitly given integral transform of H. (Actually, the formula is a little more complicated: the term H̃(λ_{j}) corresponding to the conjugacy class of a hyperbolic element which is the nth 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 Γ.)

By choosing for H a suitable function depending on a complex parameter s and exponentiating both sides of (1), one obtains the Selberg zeta function, Z_{Γ}(s). The expression for Z_{Γ}(s) as a product over hyperbolic elements takes the form

Z_{Γ}(s) = ∏_{γ} ∏_{m=0}^{∞}(1 − N(γ)^{−s−m})  (ℜ(s) ≫ 0),  (2)

where γ in the first product ranges over all primitive hyperbolic elements of Γ (or equivalently, all primitive geodesics of Γ\H), while the expression for Z_{Γ}(s) as a product over eigenvalues of the Laplace operator shows that Z_{Γ}(s) has a meromorphic continuation in s and has zeros (with the appropriate multiplicities) at all the spectral parameters of Γ, defined as the numbers s_{j} with s_{j}(1 − s_{j}) = λ_{j}. (Notice that there are two s_{j} for each eigenvalue λ_{j}, the values corresponding to the two eigenvalues given above for Γ = Γ_{1} being \frac{1}{2} ± 9.53369526 ⋯ i and \frac{1}{2} ± 13.779751 ⋯ i.)

In this note we will concentrate mostly on the case of the full modular group Γ_{1} = SL_{2}(ℤ) and its order 2 extension Γ_{+} = GL_{2}(ℤ), which can be made to operate on H by letting γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) act by z ↦ (az̄ + b)/(cz̄ + d) if det(γ) = −1. The eigenvalues of the Laplace operator for Γ_{+} are just the even eigenvalues for Γ_{1}, i.e., those corresponding to eigenfunctions which are symmetric under z ↦ −z̄ (of the two eigenvalues for Γ_{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_{Γ_{+}}(s), which is defined by a formula like (2) but with γ now ranging over the primitive hyperbolic elements of Γ_{+} and with the factor 1 − N(γ)^{−s−m} replaced by 1 − (−1)^{m}N(γ)^{−s−m} whenever γ has determinant −1.

§2. The Mayer transfer operator 𝔏_{s}

A surprising new interpretation of the Selberg zeta function was found by D. Mayer [9]. Let D denote the open disc of radius 3/2 and center 1 and V the space of functions which are holomorphic in D and extend continuously to its boundary. Then Mayer defines an operator 𝔏_{s} : V → V by the formula

𝔏_{s}ϕ(z) = ∑_{n=1}^{∞} \frac{1}{(z + n)^{2s}} ϕ(\frac{1}{z + n})  (ϕ ∈ V, z ∈ D) .  (3)

The series converges for ℜ(s) > \frac{1}{2} for any ϕ ∈ V, and can be continued meromorphically to all s. (Write ϕ(z) as the sum of a polynomial of degree k and a function which is O(z^{k+1}) as z → 0; 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 ℜ(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:

Theorem. We have the identities

Z_{Γ_{1}}(s) = det(1 − 𝔏_{s}^{2}),  Z_{Γ_{+}}(s) = det(1 − 𝔏_{s}) .  (4)

(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) → [0, 1) which assigns to any x > 0 the fractional part of 1/x. For each complex number s with ℜ(s) > \frac{1}{2} and each positive integer n we then have the partition function

Z_{n}(F, h_{s}) = ∑_{x∈[0,1), F^{n}x=x} h_{s}(x)h_{s}(F x) ⋯ 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 ∈ [0, 1) are the points (n + x)^{−1} (n = 1, 2, …), 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/Γ_{+}, and using this correspondence one finds that the sub-product of (2) corresponding to m = 0 for the group Γ = Γ_{1} is equal to exp(∑_{n=1}^{∞} Z_{2n}(F, h_{s})/n). (Here only even indices occur because the map x ↦ (n+x)^{−1} has determinant −1. For Γ_{+} 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 Γ_{+}, is presented in [4].

§3. 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 Γ_{+} (i.e., an even eigenfunction for Γ_{1}) with spectral parameter s a holomorphic solution in ℂ ∖ (−∞, 0] of the three-term functional equation

ψ(z) = ψ(z + 1) + z^{−2s} ψ(1 + 1/z) .  (5)

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 realanalytic 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.

Theorem. Let s be a complex number with ℜ(s) = \frac{1}{2}. Then there is a canonical isomorphism between the space of even or odd eigenfunctions of the Laplace operator for Γ_{1} with eigenvalue s(1 − s) and the space of real-analytic functions ψ on (0, ∞) satisfying the even or odd three-term functional equation

ψ(x) = ψ(x + 1) ± x^{−2s} ψ(1 + 1/x)  (6)

and the growth condition lim_{x→∞} ψ(x) = 0.

Note that the theorem implies in particular that the space of real-analytic solutions of (6) satisfying the growth condition ψ(x) = o(1) as x → ∞ 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 ψ(x) = O(1) as x → ∞ is infinitedimensional 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 ψ(x) = ψ(x + 1) + (x + 1)^{−2s} ψ(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 Γ_{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 Eichler integral f^{∗}(z), defined most easily by the formula f^{∗}(z) = ∑_{n=1}^{∞} n^{1−k}a_{n} e^{2πinz}, where f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz} is the Fourier expansion of f; then the function ψ(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 Γ_{1} (a so-called Maass wave form) with spectral parameter s, then, as is well known, f has a Fourier expansion of the form

f(z) = \sqrt{y} ∑_{n=1}^{∞} A_{n} K_{s− \frac{1}{2}} (2πny) cos(2πnx)  (z = x + iy ∈ H)  (7)

in the even case and a similar expansion with “sin” instead of “cos” in the odd case, where K_{ν}(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 nonholomorphic, we define its “Eichler integral” to be the holomorphic function f^{∗} in the upper half-plane given by

f^{∗}(z) = ∑_{n=1}^{∞} n^{s− \frac{1}{2}} A_{n} e^{2πinz}  (z ∈ H) .  (8)

We extend this function to ℂ ∖ 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^{∗}(±iy) (y ∈ ℝ_{+}) to the L-series of f and using the functional equation of this L-series, one shows that the function ψ(z) defined in ℂ ∖ ℝ by the formula ψ(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∖ℝ to the cut plane ℂ∖(−∞, 0]. The function ψ of the theorem is then simply the restriction of this extended function to the positive real axis. That ψ satisfies the three-term functional equation in ℂ ∖ 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 ↦ 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 Γ_{1}, is the fact that ψ extends analytically across the positive real axis.

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

§4. Interrelationships and generalizations

We now consider the relationship between the various objects we have described. In §1 we saw how the length spectrum of Γ\H can be encoded in an infinite product, the Selberg zeta function, whose meromorphic continuation has zeros at the spectral parameters s_{j} of Γ. In §2 we saw how this zeta function, for Γ = Γ_{+} or Γ_{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 ∏_{ν}(1 − β_{ν}(s)) or ∏_{ν}(1 − β_{ν}(s)^{2}), where β_{ν}(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 ±1. We thus obtain as a direct corollary of the theorem in §2 the following statement: If s is the spectral parameter for an even (resp. odd) eigenfunction of the Laplace operator for SL_{2}(ℤ), then there exists a non-zero function ϕ ∈ V satisfying 𝔏_{s}ϕ = ϕ (resp. 𝔏_{s}ϕ = −ϕ).

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}ϕ(x) = (x + 1)^{−2s}ϕ(1/(x + 1)) + 𝔏_{s}ϕ(x + 1), so if ϕ ∈ V is an eigenfunction of 𝔏_{s} with eigenfunction ±1 then we have the identity ϕ(x) = ϕ(x + 1) ± (x + 1)^{−2s}ϕ(1/(x + 1)), and after the simple change of variables ϕ(x) = ψ(x+1) this is precisely equation (6).

We can therefore summarize the whole story by saying that there are very close relationships between four a priori very different objects, as indicated in the following diagram,

A. Set of lengths of geodesics on Γ_{1}\H

[figure]

[figure] B. Spectrum of the Laplace operator Δ

[figure]

[figure]

C. Spectrum of the Mayer operator 𝔏_{s} [figure] [figure]

D. Lewis’s three-term functional equation

[figure]

[figure]

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 ±1 is, up to a shift by 1, a solution of the functional equation (6).

It is also natural to ask what happens when Γ_{1} is replaced by some other discrete group Γ ⊂ SL_{2}(ℝ). The relationship between the lengths of geodesics and the eigenvalues of the Laplacian for Γ\H given by the Selberg trace formula works equally well, as discussed in §1, but both the Mayer operator and the three-term functional equation depend on the special properties of the group Γ_{1}, and specifically on the fact that it is generated by the two transformations S : z ↦ −1/z and T : z ↦ z + 1. If Γ is a subgroup of Γ_{1} of finite index N, then we can reduce to the case of Γ_{1} by replacing the functions on which 𝔏_{s} acts and the function ψ by vector-valued functions of length N, with coordinates indexed by the set of cosets Γ\Γ_{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 ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) on Γ\Γ_{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 Γ is not contained in, or even commensurable with, the modular group Γ_{1}. In the case of the Mayer operator, this requires considering the analogue of the continued fraction expansion appropriate to the group Γ, based on the geometry of a chosen fundamental domain for Γ, 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 Γ_{+} rather than to Γ_{1} (as seen by the fact that the Selberg zeta function for Γ_{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 Γ_{1}; the right modification of (3) is the operator ϕ(z) ↦ ∑_{n=2}^{∞}(n − z)^{−2s}ϕ(1/(n − z)), whose fixed points correspond via ϕ(z) = (1 − z)^{−2s}ψ(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 Γ is that of cohomology of groups. We indicate here very briefly how this works. (The details will be given in [6].)

Define V_{s}^{∞} to be the space of C^{∞} functions ψ : ℝ → ℂ such that the function |x|^{−2s}ψ(1/x) on ℝ ∖ {0} extends to a C^{∞} function on ℝ (or equivalently, that the function Ψ(x, y) := |y|^{−2s}ψ(x/y) on ℝ^{2} ∖{(0, 0)} is C^{∞}), and let V_{s}^{ω} be the subspace defined the same way but with “C^{∞}” replaced by “real-analytic.” We define an intermediate space V_{s}^{ω/2} as the space of functions in V_{s}^{∞} which are real-analytic except at a finite subset of ℙ^{1}(ℝ). The group G = SL_{2}(ℝ) acts on all three spaces by (ψ|g)(x) = |cx+d|^{−2s}ψ(g(x)) for g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ G (or equivalently, by Ψ ↦ Ψ∘g). In [5] it is shown that the function ψ occurring in the theorem in §2, if extended from ℝ_{+} to ℝ∖{(0, 0)} by ψ(x) = ∓ψ(−x) (or equivalently by ψ(x) = −|x|^{−2s}ψ(−1/x)), belongs to V_{s}^{∞} and hence also to V_{s}^{ω/2}, and now the three-term functional equation implies exactly that this extended function satisfies the equations ψ+ψ|S = ψ+ψ|U +ψ|U^{2} = 0, where U = ST is the standard element of Γ_{1} of order 3 which together with the involution S generates this group. These equations say that the map S ↦ ψ, T ↦ 0 extends to a cocycle on Γ with values in V_{s}^{ω/2}, and the fact that T maps to 0 says that this is a parabolic cocycle (one which sends every parabolic element γ ∈ Γ to an element in the image of 1 − γ). The theorem of §2 can then be reformulated as the statement that the parabolic cohomology group H_{par}^{1}(Γ_{1}, V_{s}^{ω/2}) 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 Γ is simply the same assertion for H_{par}^{1}(Γ, V_{s}^{ω/2}) and the spectral parameters for Γ. (If Γ is contained in Γ_{1} and one uses the point of view of vector-valued functions mentioned above, then one would look instead at H_{par}^{1}(Γ_{1}, (V_{s}^{ω/2})^{Γ\Γ_{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}^{ω}; these new cocycles are no longer parabolic, but satisfy the weaker “semi-parabolic” condition that their value at every parabolic element γ of Γ belongs to V_{s}^{ω/2}|(1 − γ), and the cohomology group H_{par/2}^{1}(Γ, V_{s}^{ω}) defined by such semi-parabolic real-analytic cocycles modulo coboundaries is again isomorphic to the space of Maass wave forms on Γ with spectral parameter s.

§5. 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_{Γ}(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 §4, the eigenvalues β_{ν}(s) of the operator 𝔏_{s} tend to zero rapidly, so that the infinite products ∏(1−β_{ν}(s)) and ∏(1−β_{ν}(s)^{2}) defining the Fredholm determinants det(1 − 𝔏_{s}) = Z_{Γ_{+}}(s) and det(1 − 𝔏_{s}^{2}) = Z_{Γ_{1}}(s) are rapidly convergent. Hence if we have a good algorithm to compute the eigenvalues β_{ν}(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, ∞) given by the kernel function

K_{s}(x, y) = \frac{J_{2s−1}(2\sqrt{xy})}{\sqrt{(e^{x} − 1)(e^{y} − 1)}} ,

where J_{ν}(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, …) to B_{k}(x) = ζ(x + 1, k + 2s − 1), where ζ(α, ν) = ∑_{n=1}^{∞}(n + α)^{−ν} denotes the Hurwitz zeta function. If one takes m well-chosen points x_{1}, …, x_{m} in [0, 1] (a good choice is the set of points x_{j} = sin^{2}((2j − 1)π/4m)), then it turns out that the eigenvalues of the m × m matrix A^{−1}B, where A is the Vandermonde matrix (A_{k}(x_{j}))_{1≤j,k≤m} and B the matrix (B_{k}(x_{j}))_{1≤j,k≤m}, are good approximations to the desired eigenvalues β_{ν}(s) of 𝔏_{s}. (More precisely, the matrix AB^{−1} has some eigenvalues β_{ν,m}(s) which tend exponentially rapidly to β_{ν}(s) as m → ∞, 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 β_{ν} = β_{ν}(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 β_{ν}(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.

β_{1} = 1.0000000000000000000000000000,
β_{2} = −0.3036630028987326585974481219
β_{3} = 0.1008845092931040753056376425,
β_{4} = −0.03549615902165984540889163041,
β_{5} = 0.01284379036244026481516090344,
β_{6} = −0.004717777511571031073863594144,
β_{7} = 0.001748675124305511914510617525,
β_{8} = −0.0006520208583205029031816912077,
β_{9} = 0.0002441314655245138906981706362,
β_{10} = −0.00009168908376859566896987721814
β_{11} = 0.00003451654616347253214653427704,
β_{12} = −0.00001301769787581775595712876501,
β_{13} = 0.000004916782323649260641924415470,
β_{14} = −0.000001859377799671070019642381811,
β_{15} = 0.0000007038108595990962920678071171,
β_{16} = −0.0000002666477929249816513412730701,
β_{17} = 0.00000005079864542122092292865860954,
β_{18} = −0.00000003831081550679296190236314295,
β_{19} = 0.00000001400621008979342385785713695.

Table 1. Eigenvalues of 𝔏_{1}

Applying this method with s of the form \frac{1}{2} + it with t real, we can compute the eigenvalues β_{ν}(s) to high accuracy for small values of t. If we plot the curve of the largest (in absolute value) eigenvalue β_{1}(\frac{1}{2} +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 ≈ 9.53369526135355746 and t ≈ 13.77975135189074, respectively, in accordance with Mayer’s theorem from §2 and the known numerical values of the first odd and even spectral parameters for Γ_{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 Γ_{1}, an invariant which plays a role in the work of Jorgenson and Kramer discussed at this conference.

References

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[2] I. Efrat, Dynamics of the continued fraction map and the spectral theory of SL(2, Z), Invent. Math. 114 (1993), 207–218.

[3] J.B. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms, Invent. Math. 127 (1997), 271–306.

[4] J.B. Lewis and D. Zagier, Period functions and the Selberg zeta function for the modular group, The Mathematical Beauty of Physics, Adv. Series in Math. Physics 24, World Scientific, Singapore, 1997, pp. 83–97.

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[6] J.B. Lewis and D. Zagier, Period functions for Maass wave forms. II, in preparation.

[7] Yu.I. Manin and M. Marcolli, Continued fractions, modular symbols, and non-commutative geometry, Selecta Math., to appear.

[8] F. Martin, Périodes de formes modulaires de poids 1, Thèse, Université de Paris 7 (2001).

[9] D. Mayer, The thermodynamic formalism approach to Selberg’s zeta function for P SL(2, Z), Bull. AMS 25 (1991), 55–60.