A SPECTRAL INTERPRETATION OF THE ZEROS OF THE CONSTANT TERM OF

CERTAIN EISENSTEIN SERIES

WERNER MÜLLER

Abstract. In this paper we consider the constant term φ_{K}(y, s) of the non-normalized Eisenstein series attached to PSL(2, O_{K}), where K is either ℚ or an imaginary quadratic field of class number one. The main purpose of this paper is to show that for every a ≥ 1 the zeros of the Dirichlet series φ_{K}(a, s) admit a spectral interpretation in terms of eigenvalues of a natural self-adjoint operator Δ_{a}. This implies that, except for at most two real zeros, all zeros of φ_{K}(a, s) are on the critical line, and all zeros are simple. For K = ℚ this is due to Lagarias and Suzuki [LS] and Ki [Ki].

0. Introduction

Let Γ = PSL(2, ℤ) be the modular group and ℍ the upper half-plane. We consider the (completed) nonholomorphic Eisenstein series E^{∗}(z, s) for the modular group which is given for z ∈ ℍ and Re(s) > 1 by

(0.1)  E^{∗}(z, s) := π^{−s}Γ(s) (\frac{1}{2} ∑_{(m,n)∈ℤ^{2}\(0,0)} \frac{y^{s}}{|mz + n|^{2s}} ) .

It is well known that for fixed z, E^{∗}(z, s) admits a meromorphic continuation to the whole s-plane, and satisfies the functional equation

E^{∗}(z, s) = E^{∗}(z, 1 − s).

Its only singularities are simple poles at s = 1 and s = 0. As a function of z, it is invariant under Γ

E^{∗}(γz, s) = E^{∗}(z, s), γ ∈ Γ.

In particular E^{∗}(z, s) is invariant under z ↦ z + 1 and so it has a Fourier expansion

E^{∗}(x + iy, s) = ∑_{n∈ℤ}a_{n}(y, s)e^{2πix}.

Date: March 13, 2007. 1991 Mathematics Subject Classification. Primary: 11M36; Secondary: 11M26. Key words and phrases. Eisenstein series, zeros of Dirichlet series, cut-off Laplacian.

The zeroth Fourier coefficient φ_{0}(y, s) is given by

φ_{0}(y, s) = Λ(2s)y^{s} + Λ(2s − 1)y^{1−s},

where Λ(s) is the completed zeta function

Λ(s) = π^{−s/2}Γ (\frac{s}{2}) ζ(s).

Let a > 0. The zeros of φ_{0}(a, s) have been studied by various authors. Hejhal [He, Proposition 5.3(f)] proved that for all a ≥ 1, the complex zeros of φ_{0}(a, s) are on the critical line Re(s) = 1/2. Lagarias and Suzuki [LS] reproved this result and also determined the occurrence of real zeros. Ki [Ki] proved that all complex zeros are simple. Putting these results together, we have following theorem.

Theorem 0.1. For each a ≥ 1 all complex zeros of φ_{0}(y, s) are simple and lie on the critical line Re(s) = 1/2. Moreover there is a critical value a^{∗} = 4πe^{−γ} = 7.055... such that the following holds:

1) For 1 ≤ a ≤ a^{∗} all zeros are on the critical line.

2) For a > a^{∗} there are exactly two zeros off the critical line. These are real simple zeros ρ_{a}, 1 − ρ_{a} with ρ_{a} ∈ (1/2, 1). The zero ρ_{a} is a nondecreasing function of a and lim_{a→∞} ρ_{a} = 1.

The first aim of this paper is to point out that there is actually a spectral interpretation of the zeros of φ_{0}(a, s), which gives another proof of this theorem. For a > 0 let Δ_{a} be the cut-off Laplacian Δ_{a} introduced by Lax and Phillips [LP]. It acts in the subspace H_{a} ⊂ L^{2}(Γ\ℍ) of all f ∈ L^{2}(Γ\ℍ) satisfying ∫_{0}^{1} f(x + iy) dx = 0 for almost all y ≥ a. The cut-off Laplacian Δ_{a} is a nonnegative self-adjoint operator with pure point spectrum. The spectrum has been studied by Colin de Verdiere [CV]. Let

(0.2)  c(s) = \frac{Λ(2s − 1)}{Λ(2s)}, s ∈ ℂ.

The following theorem is an immediate consequence of [CV, Théorème 5].

Theorem 0.2. For every a > 0, the spectrum of Δ_{a} is the union of the cuspidal eigenvalues 0 < λ_{1} ≤ λ_{2} ≤ ⋯ → ∞ of Δ and a sequence of eigenvalues

0 < μ_{0}(a) < μ_{1}(a) < ⋯

with the following properties:

1) Each eigenvalue μ_{j}(a) is a decreasing function of a.

2) If a ≥ 1, each eigenvalue μ_{j}(a) has multiplicity 1. Moreover lim_{a→∞} μ_{0}(a) = 0 and if j ≥ 1, then μ_{j}(a) ≥ 1/4 and lim_{a→∞} λ_{j}(a) = 1/4.

3) Let a ≥ 1. Then the map s ↦ s(1 − s) is a bijection between the zeros ρ ≠ 1/2 of φ_{0}(a, s) and the eigenvalues μ_{j}(a) ≠ 1/4 of Δ_{a}.

4) 1/2 is a zero of φ_{0}(a, s) for all a > 0. There is j with μ_{j}(a) = 1/4 if and only if c′(1/2) = −2 log a.

This theorem implies that for a ≥ 1 there is at most one zero of φ_{0}(a, s) which is off the line Re(s) = 1/2. The simplicity of the zeros follows from the consideration of the corresponding eigenfunctions of Δ_{a}.

The main purpose of this article is to extend these results to the constant term of other Eisenstein series. In the present paper we will consider the Eisenstein series attached to PSL(2, O_{K}), where O_{K} is the ring of integers of an imaginary quadratic field K = ℚ(\sqrt{−D}) of class number one, D being a square free positive integer. These are exactly the fields ℚ(\sqrt{−D}) with D = 1, 2, 3, 7, 11, 19, 43, 67, 163 [St]. Let d_{K} be the discriminant of K and let ζ_{K}(s) be the Dedekind zeta function of K. Let

(0.3)  Λ_{K}(s) = (\frac{\sqrt{|d_{K}|}}{2π})^{s} Γ(s)ζ_{K}(s)

be the completed zeta function. Then Λ_{K}(s) satisfies the functional equation Λ_{K}(s) = Λ_{K}(1 − s). For a > 0 let

(0.4)  φ_{K}(a, s) := a^{s}Λ_{K}(s) + a^{2−s}Λ_{K}(s − 1).

Note that φ_{K}(a, s) is a Dirichlet series which satisfies the functional equation φ_{K}(a, s) = φ_{K}(a, 2 − s). Let ξ_{K}(s) = s(s − 1)Λ_{K}(s). Then ξ_{K}(s) is entire. Put

a_{K}^{∗} := exp (1 + \frac{ξ′_{K}(1)}{ξ_{K}(1)}) .

Then our main result is the following theorem.

Theorem 0.3. Let K = ℚ(\sqrt{−D}) be an imaginary quadratic field of class number one. Then for each a ≥ 1 all complex zeros of φ_{K}(a, s) are simple and lie on the line Re(s) = 1. Moreover

1) For a > max{a_{K}^{∗}, 1} there are exactly two zeros off the critical line. These are simple zeros ρ_{a}, 2 − ρ_{a} with ρ_{a} ∈ (1, 2). The zero ρ_{a} is a nondecreasing function of a, and lim_{a→∞} ρ_{a} = 2.

2) If 1 ≤ a_{K}^{∗} and 1 ≤ a ≤ a_{K}^{∗}, all zeros of φ_{K}(a, s) are on the critical line.

To prove Theorem 0.3, we extend Colin de Verdiere’s Theorem [CV, Théorème 5] to our setting. Let O_{K} be the ring of integers of K and let Γ = PSL(2, O_{K}). Then Γ is a discrete subgroup of PSL(2, ℂ) which acts properly discontinuously on the 3-dimensional hyperbolic space ℍ^{3}. The quotient Γ\ℍ^{3} has finite volume and a single cups at ∞. The function φ_{K}(y, s) appears to be the zeroth Fourier coefficient of the modified Eisenstein series attached to the cusp κ = ∞. Let Δ_{a} be the corresponding cut-off Laplacian on Γ\ℍ^{3}. Then we generalize Theorem 0.2 to this setting. As above, this leads to a spectral interpretation of the zeros of φ_{K}(a, s), and we deduce Theorem 0.3 from this spectral interpretation.

The constant a_{K}^{∗} can be computed using the Kronecker limit formula [La, p. 273] and the Chowla-Selberg formula [SC, (2), p.110]. For example, we get

a_{ℚ(i)}^{∗} = \frac{4π^{2}e^{2+γ}}{Γ(1/4)^{4}} ≈ 3.00681, a_{ℚ(\sqrt{−2})}^{∗} = \frac{8π^{2}e^{2+γ}}{(Γ (\frac{1}{8}) Γ (\frac{3}{8}))^{2}} ≈ 3.2581.

Thus a_{ℚ(\sqrt{−D})}^{∗} > 1 for D = 1, 2 and therefore, in the range 1 ≤ a ≤ a_{ℚ(\sqrt{−D})}^{∗} all zeros of φ_{ℚ(\sqrt{−D})}(a, s) are on the line Re(s) = 1.

The method used to prove Theorem 0.3 can be extended so that an arbitrary number field K of class number one can be treated. The underlying global Riemannian symmetric space is X = ℍ^{r_{1}} × (ℍ^{3})^{r_{2}}, where r_{1} is the number of real places and r_{2} the number of pairs of complex conjugate places of K. The structure of Γ\X is slightly more complicated, however the proof of the corresponding statement is completely analogous. More generally, it seems to be possible to prove similar results for the constant terms of rank one cuspidal Eisenstein series attached to PSL(n, ℤ).

1. The cut-off Laplacian

The cut-off Laplacian can be defined for every rank one locally symmetric space. In the present paper we are dealing only with hyperbolic manifolds of dimension 2 and 3. So we discuss only the case of a hyperbolic 3-manifold. The general case is similar.

Let

ℍ^{3} = {(x_{1}, x_{2}, y) ∈ ℝ^{3} : y > 0}

be the hyperbolic 3-space with its hyperbolic metric

ds^{2} = \frac{dx_{1}^{2} + dx_{2}^{2} + dy^{2}}{y^{2}} .

The hyperbolic Laplacian Δ is given by

Δ = −y^{2} (\frac{∂^{2}}{∂x_{1}^{2}} + \frac{∂^{2}}{∂x_{2}^{2}} + \frac{∂^{2}}{∂y^{2}} ) + y \frac{∂}{∂y} .

If we regard ℍ^{3} as the set of all quaternions z+yj with z ∈ ℂ and y > 0, then G = PSL(2, ℂ) is the group of all orientation preserving isometries of ℍ^{3}. It acts by linear fractional

transformations, i.e., for γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ G,

γ(w) = (aw + b)(cw + d)^{−1}, w ∈ ℍ^{3}.

Let

B(ℂ) := \{(\begin{matrix}λ & z \\ 0 & λ^{−1} \end{matrix}) : λ ∈ ℂ^{∗}, z ∈ ℂ\} /\{± Id\}, N(ℂ) := \{(\begin{matrix}1 & z \\ 0 & 1 \end{matrix}) : z ∈ ℂ\} .

Let Γ ⊂ G be a discrete subgroup of finite co-volume. Let κ_{1} = ∞, κ_{2}, ..., κ_{m} ∈ ℙ^{1}(ℂ) be a complete set of Γ-inequivalent cusps, and let Γ_{i} be the stabilizer of κ_{i} in Γ. Choose σ_{i} ∈ G such that σ_{i}(κ_{i}) = ∞, i = 1, ..., m. Then

(1.1)  σ_{i}Γ_{i}σ_{i}^{−1} ∩ N(ℂ) = \{(\begin{matrix}1 & λ \\ 0 & 1 \end{matrix}) : λ ∈ L_{i}\} ,

where L_{i} is a lattice in ℂ [EGM, Theorem 2.1.8]. Note that for D ≠ 1, 3 the intersection actually coincides with σ_{i}Γ_{i}σ_{i}^{−1}. Choose closed fundamental sets P_{i} for the action of σ_{i}Γ_{i}σ_{i}^{−1} on ℙ^{1}(ℂ) \ {∞} = ℂ. For T > 0 define

F̃_{i}(T) := {(z, y) ∈ ℍ^{3} : z ∈ P_{i}, y ≥ T } .

Let F_{i}(T) = σ_{i}^{−1}(F̃_{i}(T)). There exists T_{i} > 0 such that any two points in F_{i}(T_{i}) are Γequivalent if and only if they are Γ_{i}-equivalent. For such a choice of T_{1}, ..., T_{m} there exists a compact subset F_{0} ⊂ ℍ^{3} such that

(1.2)  F := F_{0} ∪ F_{1}(T_{1}) ∪ ⋯ ∪ F_{m}(T_{m})

is a fundamental domain for Γ [EGM, Proposition 2.3.9]. Let

(1.3)  b := max{T_{1}, ..., T_{m}}.

By C_{c}^{∞}(Γ\ℍ^{3}) we denote the space of Γ-invariant C^{∞}-functions on ℍ^{3} with compact support in F. For f ∈ C_{c}^{∞}(Γ\ℍ^{3}) let

∥ f ∥^{2}= ∫_{F} |f(x_{1}, x_{2}, y)|^{2} \frac{dx_{1}dx_{2}dy}{y^{3}} ,

and let L^{2}(Γ\ℍ^{3}) be the completion of C_{c}^{∞}(Γ\ℍ^{3}) with respect to this norm. Similarly let

∥ df ∥^{2}= ∫_{F} df ∧ ∗\overline{df} = ∫_{F} ∥ df(x_{1}, x_{2}, y) ∥^{2} \frac{dx_{1}dx_{2}dy}{y} .

Let H^{1}(Γ\ℍ^{3}) denote the completion of C_{c}^{∞}(Γ\ℍ^{3}) with respect to the norm

∥ f ∥_{1}^{2}:=∥ f ∥^{2} + ∥ df ∥^{2} .

Denote by |P_{i}| the Euclidean area of the fundamental domain P_{i} ⊂ ℂ of σ_{i}Γ_{i}σ_{i}^{−1}. For f ∈ L^{2}(Γ\ℍ^{3}) let

f_{j,0}(y) = \frac{1}{|P_{i}|} ∫_{P_{i}} f(σ_{i}^{−1}(x_{1}, x_{2}, y)) dx_{1}dx_{2}

be the zeroth coefficient of the Fourier expansion of f at the cusp κ_{j}. Note that for f ∈ H^{1}(Γ\ℍ^{3}), each f_{j,0} belongs to H^{1}(ℝ^{+}) and therefore, each f_{j,0} is a continuous function on ℝ^{+}. For a > 0 let

H_{a}^{1}(Γ\ℍ^{3}) := {f ∈ H^{1}(Γ\ℍ^{3}): f_{j,0}(y) = 0 for y ≥ a, j = 1, ..., m} .

Then H_{a}^{1}(Γ\ℍ^{3}) is a closed subspace of H^{1}(Γ\ℍ^{3}). Hence the quadratic form

q_{a}(f) =∥ df ∥^{2}, f ∈ H_{a}^{1}(Γ\ℍ^{3}),

is closed. Let Δ_{a} denote the self-adjoint operator which represents the quadratic form q_{a}. It acts in the Hilbert space H_{a} which is the closure of H_{a}^{1}(Γ\ℍ^{3}) in L^{2}(Γ\ℍ^{3}). By definition, Δ_{a} is nonnegative. Its domain can be described as follows. Let ψ_{j,a} ∈ D′(Γ\ℍ^{3}) be defined by

ψ_{j,a}(f) := f_{j,0}(a), f ∈ C_{c}^{∞}(Γ\ℍ^{3}).

Let b > 0 be defined by (1.3).

Lemma 1.1. Let a ≥ b. Then the domain of Δ_{a} consists of all f ∈ H_{a}^{1}(Γ\ℍ^{3}) for which there exist C_{1}, ..., C_{m} ∈ ℂ such that

(1.4)  Δf − ∑_{j=1}^{m} C_{j}ψ_{j,a} ∈ L^{2}(Γ\ℍ^{3}).

The proof of the lemma is analogous to the proof of Théorèm 1 in [CV]. Let f ∈ dom(Δ_{a}). By the lemma, there exist C_{1}, ..., C_{m} ∈ ℂ such that (1.4) holds. Then Δ_{a}f is given by

(1.5)  Δ_{a}f = Δf − ∑_{j=1}^{m} C_{j}ψ_{j,a}.

Furthermore, we have

Lemma 1.2. Δ_{a} has a compact resolvent.

Proof. The proof is a simple extension of the argument in [LP, p.206]. □

So the spectrum of Δ_{a} consists of a sequence of eigenvalues 0 ≤ λ_{1}(a) ≤ λ_{2}(a) ≤ ⋯ → ∞ with finite multiplicities. To describe the eigenvalues and eigenfunctions of Δ_{a} more explicitely, we need to consider the Eisenstein series. Let σ_{j} ∈ G, j = 1, ..., m, be as above. For each cusp κ_{i}, the Eisenstein series E_{i}(w, s) attached to κ_{i} is defined to be

E_{i}(w, s) := ∑_{γ∈Γ_{i}\Γ} y(σ_{i}γw)^{s}, Re(s) > 2,

where y(σ_{i}γw) denotes the y-component of σ_{i}γw. Selberg has shown that E_{i}(w, s) can be meromorphically continued in s to the whole complex plane ℂ, and it is an automorphic eigenfunction of Δ with

ΔE_{i}(w, s) = s(2 − s)E_{i}(w, s).

Since E_{i}(w, s) is Γ-invariant, it is invariant under the stabilizer Γ_{j} of the cusp κ_{j} and therefore, admits a Fourier expansion at κ_{j}. The constant Fourier coefficient is given by

(1.6)  δ_{ij}y^{s} + C_{ij}(s)y^{2−s}.

Put

C(s) := (C_{ij}(s))_{i,j=1,...,m} .

This is the so called “scattering matrix”. The scattering matrix and the Eisenstein series satisfy the following system of functional equations.

\begin{matrix}C(s)C(2 − s) = Id, \\ E_{i}(w, s) = ∑_{j=1}^{m} C_{ij}(s)E_{j}(w, 2 − s), i = 1, ..., m. \end{matrix} ; (1.7)

Now recall that a square integrable eigenfunction f of Δ is called cuspidal, if the zeroth Fourier coefficient f_{j,0} of f at the cusp κ_{j} vanishes for all j = 1, ..., m. Denote by S(λ; Γ) the space of cuspidal eigenfunctions of Δ with eigenvalue λ. A function f ∈ C^{0}(Γ\ℍ^{3}) is called of moderate growth, if there exists R ∈ ℝ^{+} such that

f(σ_{i}^{−1}(x_{1}, x_{2}, y)) = O(y^{R}), for (x_{1}, x_{2}, y) ∈ F_{i}(b), i = 1, ..., m.

It follows from the Fourier expansion in the cusps [EGM, Section 3.3.3] that a cuspidal eigenfunction f of Δ is rapidly decreasing in each cusp. Therefore for ψ ∈ E(λ, Γ) and φ ∈ S(λ, Γ) , the inner product ⟨ψ, φ⟩ is well defined. For λ ∈ ℂ let

(1.8) E(λ; Γ) := {ψ ∈ L2_{loc}(Γ\ℍ^{3}): Δψ = λψ, ψ is of moderate growth, ψ ⊥ S(λ; Γ)} .

Lemma 1.3. Let m be the number of cusps of Γ\ℍ^{3}. Then

1) For every λ ∈ ℂ we have dim E(λ; Γ) ≤ m.

2) Suppose that λ = s(2 − s), where Re(s) ≥ 1, s ≠ 1, and s is not a pole of any Eisenstein series. Then E_{1}(w, s), ..., E_{m}(w, s) is a basis of E(λ; Γ).

Proof. The proof is analogous to the proof of Satz 11 in [Ma, p.171]. Let ϕ, ψ ∈ E(λ; Γ) and λ = s(2 − s). The zeroth Fourier coefficient of ϕ, ψ at the j-th cusp is given by

ϕ_{0,j}(y) = a_{j}y^{s} + b_{j}y^{2−s}, ψ_{0,j}(y) = c_{j}y^{s} + d_{j}y^{2−s}, s ≠ 1,

and for s = 1 by

ϕ_{0,j}(y) = a_{j}y + b_{j}y log y, ψ_{0,j}(y) = c_{j}y + d_{j}y log y.

Let χ_{j,[a,∞)} denote the characteristic function of Γ_{i}\F_{i}(a) in Γ\ℍ^{3}. Let

ϕ_{a} = ϕ − ∑_{j=1}^{m} χ_{j,[a,∞)}ϕ_{0,j}, ψ_{a} = ψ − ∑_{j=1}^{m} χ_{j,[a,∞)}ψ_{0,j}

be the truncations of ϕ and ψ, respectively, at level a ≥ b. Let s ≠ 1. Integrating by parts, we get

\begin{aligned}0 = ∫_{Γ\ℍ^{3}}(ϕ_{a}Δψ_{a} − ψ_{a}Δϕ_{a}) dvol & = ∑_{j=1}^{m} (ψ_{0,j}(y)\frac{∂ϕ_{0,j}}{∂y} (y) − ϕ_{0,j}(y)\frac{∂ψ_{0,j}}{∂y} (y)) |_{y=a} \\ & = (2 − 2s) ∑_{j=1}^{m}(a_{j}d_{j} − b_{j}c_{j}). \end{aligned} ; (1.9)

For s = 1 the right hand side equals ∑_{j=1}^{m}(a_{j}d_{j} − b_{j}c_{j}).

An element of E(λ) is uniquely determined by its zeroth Fourier coefficients. Define i: E(λ) → ℂ^{2m} by i(ϕ) := (a_{1}, b_{1}, ..., a_{m}, b_{m}), where a_{j}, b_{j} are the 0-th Fourier coefficients of ϕ at the j-th cusp. Then i is an embedding. Let

I(x, y) := ∑_{j=1}^{m}(x_{2j−1}y_{2j} − x_{2j}y_{2j−1})

be the standard symplectic form on ℂ^{2m}. Then by (1.9) and the corresponding statement for s = 1, E(λ) is an isotropic subspace for I. Hence dim E(λ) ≤ m.

If s ≠ 1 and s is not a pole of E_{j}(w, s), j = 1, ..., m, the Eisenstein series have linearly independent 0-th Fourier coefficients given by (1.6). This implies that E_{1}(w, s), ..., E_{m}(w, s) are linearly independent and hence, form a basis of E(λ, Γ), where λ = s(2 − s). □

We are now ready to describe the spectrum of Δ_{a}. Let 0 = Λ_{0} < Λ_{1} < ⋯ < Λ_{N} < 1 be the eigenvalues of Δ with non-cuspidal eigenfunctions. Then Λ_{j} has the form Λ_{j} = σ_{j}(2 − σ_{j}) with σ_{j} ∈ (1, 2] and σ_{j} is a simple pole of some Eisenstein series E_{k}(w, s). The corresponding residue is an eigenfunction of Δ with eigenvalue Λ_{j} [EGM, Proposition 6.2.2]. The following theorem is an extension of Theorem 5 in [CV] to the 3-dimensional case.

Theorem 1.4. The spectrum of Δ_{a} is the union of the cuspidal eigenvalues (λ_{i})_{i} of Δ and a sequence 0 < μ_{0}(a) ≤ μ_{1}(a) ≤ ⋯ with the following properties:

1) Each μ_{j}(a) is a decreasing function of a,

2) Let a ≥ b. Then each μ_{j}(a) has multiplicity ≤ m, and the map s ↦ s(2 − s) is a bijection between the eigenvalues μ_{j}(a) ∉ {1, Λ_{0}, ..., Λ_{N}} and the zeros of ρ ∈ ℂ \ {1, σ_{0}, ..., σ_{N}} of φ(s) := det(C(s) + a^{2s−2} Id).

Proof. Let f ∈ L^{2}(Γ\ℍ^{3}) be a cuspidal eigenfunction of Δ. Then f belongs to H_{a}^{1}(Γ\ℍ^{3}) and Δf is square integrable. Hence by (1.4) and (1.5), f ∈ dom(Δ_{a}) and Δ_{a}f = Δf. Thus all cuspidal eigenfunctions of Δ are also eigenfunctions of Δ_{a} with the same eigenvalues.

For 1) observe that for a ≤ a′ we have

H_{a}^{1}(Γ\ℍ^{3}) ⊂ H_{a′}^{1}(Γ\ℍ^{3}).

Since Δ_{a} is a non-negative self-adjoint operator with compact resolvent, 1) follows from the mini-max characterization of its eigenvalues.

2) Let Φ be an eigenfunction of Δ_{a} with eigenvalue μ_{j}(a) = s_{j}(2 − s_{j}) and suppose that Φ ⊥ S(μ_{j}(a); Γ). Let Φ_{i,0} be the zeroth Fourier coefficient of Φ at the cusp κ_{i}. There exists 1 ≤ j ≤ m such that Φ_{j,0} ≢ 0. Let Φ̂_{i,0} denote the analytic continuation of Φ_{i,0} from (0, a) to ℝ^{+}. Put

Φ̂(w) := Φ(w) + ∑_{i=1}^{m} χ_{i,[a,∞)}Φ̂_{i,0}.

Then Φ̂ ∈ C^{∞}(Γ\ℍ^{3}) and ΔΦ̂ = μ_{j}(a)Φ̂. Moreover Φ̂ is of moderate growth and Φ̂ ⊥ S(μ_{j}(a), Γ). Therefore Φ̂ ∈ E(μ_{j}(a); Γ). By Lemma 1.3, 1), it follows that the multiplicity of μ_{j}(a) is bounded by m.

Next suppose that μ_{j}(a) = s_{j}(2 − s_{j}) ∉ {1, Λ_{0}, ..., Λ_{N}}. We can assume that Re(s_{j}) ≥ 1. As explained above, s_{j} is not a pole of any Eisenstein series E_{k}(w, s), k = 1, ..., m. Therefore, by Lemma 1.3 there exist c_{1}, ..., c_{m} ∈ ℂ such that

Φ̂(w) = ∑_{i=1}^{m} c_{i}E_{i}(w, s_{j}).

By construction, the zeroth Fourier coefficient Φ̂_{l,0}(y) of Φ̂(w) at κ_{l} vanishes at y = a for all l = 1, ..., m. By (1.6) this implies

∑_{i=1}^{m} c_{i} (δ_{il}a^{s_{j}} + C_{il}(s_{j})a^{2−s_{j}}) = 0, l = 1, ..., m.

Hence det(C(s_{j}) + a^{2s_{j}−2} Id) = 0.

Now assume that det(C(s_{j}) + a^{2s_{j}−2} Id) = 0, Re(s_{j}) ≥ 1, and s_{j} ∉ {1, σ_{0}, ..., σ_{N}}. Then s_{j} is not a pole of any Eisenstein series. Let 0 ≠ ψ ∈ ℂ^{m} such that

(1.10)  C(s_{j})ψ = −a^{2s_{j}−2}ψ.

Let ψ = (c_{1}, ..., c_{m}) and set

Φ̂(w) := ∑_{k=1}^{m} c_{k}E_{k}(w, s_{j}).

By (1.6) and (1.10), the 0-th Fourier coefficient of Φ̂(w) at the cusp κ_{l} is given by

(1.11)  ∑_{k=1}^{m} c_{k} (δ_{kl}y^{s_{j}} + C_{kl}(s_{j})y^{2−s_{j}}) = c_{l} (y^{s_{j}} − a^{2s_{j}−2}y^{2−s_{j}}) .

Therefore, the 0-th Fourier coefficient Φ̂_{l,0}(y) vanishes at y = a for all l = 1, ..., m. Put

Φ(w) = Φ̂(w) − ∑_{i=1}^{m} χ_{i,[a,∞)}Φ̂_{l,0}(y).

Then Φ ∈ H_{a}^{1}(Γ\ℍ) and it follows from the description of the domain of Δ_{a} that Φ ∈ dom(Δ_{a}) and Δ_{a}Φ = s_{j}(2 − s_{j})Φ. □

Now assume that Γ\ℍ^{3} has a single cusp κ = ∞. Then there is a single Eisenstein series E(w, s), which is given by

E(w, s) = ∑_{γ∈Γ_{∞}\Γ} y(γw)^{s}, Re(s) > 2.

The zeroth Fourier coefficient of E(w, s) at ∞ equals

(1.12)  φ_{0}(y, s) := y^{s} + c(s)y^{2−s},

where c(s) is a meromorphic function of s ∈ ℂ. The following facts are well known [EGM, pp.243-245] . The poles of E(w, s) in the half-plane Re(s) > 1 are contained in the interval (1, 2] and are all simple. The residue of E(w, s) at a pole σ ∈ (1, 2] is a square integrable eigenfunction of Δ with eigenvalue σ(2 − σ), which is non-cuspidal. Moreover all noncuspidal eigenfunctions of Δ are obtained in this way. Thus the non-cuspidal eigenvalues have multiplicity one. So in this case we get the following refinement of Theorem 1.4.

Theorem 1.5. Assume that Γ\ℍ^{3} has a single cusp. Then the spectrum of Δ_{a} is the union of the cuspidal eigenvalues (λ_{i})_{i} of Δ and a sequence 0 < μ_{0}(a) < μ_{1}(a) < ⋯ with the following properties:

1) Each μ_{j}(a) is a decreasing function of a;

2) Let a ≥ b. Then each μ_{j}(a) has multiplicity 1 and the map s ↦ s(2 − s) is a bijection between the zeros ρ ≠ 1 of φ_{0}(a, s) and the eigenvalues μ_{j}(a) ≠ 1 of Δ_{a}.

3) There is j with μ_{j}(a) = 1 if and only if c(1) = −1 and c′(1) = −2 log a.

4) Let 0 = Λ_{0} < Λ_{1} < ⋯ < Λ_{N} < 1 be the eigenvalues of Δ with non-cuspidal eigenfunctions. If a ≥ b, then

0 = Λ_{0} < μ_{0}(a) < Λ_{1} < μ_{1}(a) < ⋯ < Λ_{N} < μ_{N}(a).

Moreover,

(1.13)  lim_{a→∞} μ_{j}(a) = \begin{cases}Λ_{j} & , 0 ≤ j ≤ N; \\ 1 & , j > N. \end{cases}

Proof. 1) follows from Theorem 1.4. If s ∉ {1, σ_{0}, ..., σ_{N}}, then 2) follows also from 2) of Theorem 1.4. Now suppose that s_{0} is a pole of E(w, s) in the half-plane Re(s) ≥ 1. Then the residue ψ of E(w, s) at s_{0} is an eigenfunction of Δ with non-vanishing 0-th Fourier coefficient. Moreover ψ ⊥ S(λ, Γ). Therefore ψ ∈ E(λ, Γ), where λ = s_{0}(2 − s_{0}). By Lemma 1.3 it follows that dim E(λ, Γ) = 1. Moreover the constant term ψ_{0}(y) of ψ has the form ψ_{0}(y) = cy^{2−s_{0}}. Especially, it never vanishes on ℝ^{+}. On the other hand, if Φ were an eigenfunction of Δ_{a} with eigenvalue λ, then the corresponding eigenfunction Φ̂ ∈ E(λ, Γ), constructed in the proof of Theorem 1.4 would have a constant term which vanishes at y = a. Hence the eigenvalues Λ_{j} can not be eigenvalues of Δ_{a}. This proves 2)

For 3) we note that c(1)^{2} = 1 by the functional equation (1.7). Thus c(1) = ±1. If c(1) = −1, then E(w, 1) ≡ 0. Put ψ(w) := \frac{d}{ds}E(w, s)|_{s=1}. Then Δψ = ψ and the zeroth Fourier coefficient of ψ is given by

ψ_{0}(y) = (2 log y + c′(1))y.

Suppose that c′(1) = −2 log a. Put ψ̂_{a}(w) := ψ(w) − χ_{[a,∞)}ψ_{0}(y(w)). Then ψ̂_{a} is in the domain of Δ_{a} and Δ_{a}ψ̂_{a} = ψ̂_{a}.

For the other direction suppose that ψ̂ is an eigenfunction of Δ_{a} with eigenvalue 1 and ψ̂ ⊥ S(1, Γ). Let ψ̂_{0}(y) be the 0-th Fourier coefficient of ψ̂. Extend ψ̂_{0}(y) in the obvious way from (b, a) to a smooth function ψ_{0} on (b, ∞). Set

ψ := ψ̂ + χ_{[}a, ∞)ψ_{0}.

Then ψ is smooth, of moderate growth, satisfies Δψ = ψ and ψ ⊥ S(1, Γ). Thus ψ ∈ E(1, Γ). Therefore by Lemma 1.3 it follows that dim E(1, Γ) = 1. On the other hand, we have 0 ≠ \frac{d}{ds}E(s, w)|_{s=1} ∈ E(1, Γ). Therefore it follows that there exists c ≠ 0 such that ψ(w) = c\frac{d}{ds}E(w, s)|_{s=1}. Comparing the constant terms, it follows that c′(1) = −2 log a.

4) follows from the mini-max principle and the fact that ∪_{a≥b}H_{a}^{1}(Γ\ℍ^{3}) is dense in H^{1}(Γ\ℍ^{3}). □

2. Hyperbolic surfaces of finite area

In this section we prove Theorem 0.2 and deduce Theorem 0.1 from it.

Let Γ = PSL(2, ℤ) be the modular group. Then Γ\ℍ has a single cusp κ = ∞. As fundamental domain we take the standard domain

F := {z ∈ ℍ: | Re(z)| < 1/2, |z| > 1} .

So we can take b = 1. The Eisenstein series attached to the cusp ∞ is given by

E(z, s) = ∑_{γ∈Γ_{∞}\Γ} Im(γz)^{s} = ∑_{(m,n)=1} \frac{y^{s}}{|mz + n|^{2s}} , Re(s) > 1.

The constant term of E(z, s) equals

(2.1)  y^{s} + c(s)y^{1−s},

where c(s) is the meromorphic function defined by (0.2).

Proof of Theorem 0.2: Since the the zeros of the completed zeta function Λ(s) are all contained in the strip 0 < Re(s) < 1, it follows that Λ(2s − 1) and Λ(2s) have no common zeros. This implies that the zeros of

φ_{0}(a, s) = Λ(2s)a^{s} + Λ(2s − 1)a^{1−s}

coincide with the zeros of a^{s} +c(s)a^{1−s}. Now note that Λ(s) has poles of order 1 at s = 1, 0. The residue at s = 1 is 1 and the residue at s = 0 is −1. Hence

(2.2)  c(1/2) = lim_{s→1/2} \frac{Λ(2s − 1)}{Λ(2s)} = −1.

Now the statements 1), 3) and 4) follow immediately from [CV, Théorèm 5]. For 2) we use that by Roelcke [Ro], the smallest positive eigenvalue λ_{1} of Δ on Γ\ℍ satisfies λ_{1} > 1/4. Then 2) follows from Théorèm 5, (iii), of [CV].

Proof of Theorem 0.1: Let a ≥ 1 and let ρ ≠ 1/2 be a zero of φ_{0}(a, s). Then by Theorem 0.2, 3), ρ(1 − ρ) is an eigenvalue of Δ_{a}. Hence ρ(1 − ρ) is a non-negative real number. If ρ(1 − ρ) > 1/4, then ρ is a complex zero with Re(ρ) = 1/2. By Theorem 0.2, 2), there exist at most two zeros, ρ_{a} with 0 < ρ_{a}(1 − ρ_{a}) < 1/4. Thus ρ_{a} and 1 − ρ_{a} are zeros and we may assume that ρ_{a} ∈ (1/2, 1). Let a < a′. Then by 1), we have

−(ρ_{a} − 1/2)^{2} = ρ_{a}(1 − ρ_{a}) − 1/4 ≥ ρ_{a′}(1 − ρ_{a′}) − 1/4 = −(ρ_{a′} − 1/2)^{2}.

Thus ρ_{a} ≤ ρ_{a^{∗}} and by Theorem 0.2, 2), it follows that lim_{a→∞} ρ_{a} = 1. This proves the second part of 2).

Now consider the eigenvalue 1/4. Let

ξ(s) = s(s − 1)Λ(s).

Then we get

c′(1/2) = −\frac{c′(1/2)}{c(1/2)} = −4 lim_{s→0} (\frac{Λ′(s)}{Λ(s)} + \frac{1}{s}) = −4 (1 + \frac{ξ′(0)}{ξ(0)}) .

By [Da, pp.80-81] it follows that

2 (1 + \frac{ξ′(0)}{ξ(0)}) = −γ + log 4π,

where γ denotes Euler’s constant. Let a^{∗} := 4πe^{−γ}. Then we get

(2.3)  c′(1/2) = −2 log a^{∗}.

By Theorem 0.2, 4), 1/4 is an eigenvalue of Δ_{a} if and only if a = a^{∗}. Since all eigenvalues μ_{i}(a) of Δ_{a} are non-increasing functions of a and have multiplicity one, it follows that μ_{0}(a^{∗}) = 1/4, μ_{0}(a) < 1/4, if a > a^{∗}, and μ_{0}(a) > 1/4, if 1 ≤ a < a^{∗}. Since μ_{0}(a) = ρ_{a}(1 − ρ_{a}), this proves 1) and the first part of 2).

It remains to show that all zeros ρ ≠ 1/4 are simple. Let ρ be a zeros of φ_{0}(a, s) of multiplicity ≥ 2. Then ρ is also a zero of 1 + c(s)a^{1−2s} of multiplicity ≥ 2. Thus

0 = \frac{d}{ds} (1 + c(s)a^{1−2s}) |_{s=ρ} = a^{1−2s} (c′(ρ) − 2c(ρ) log a) .

and hence, we get

(2.4)  2 log a − \frac{c′}{c} (ρ) = 0.

Let χ_{[a,∞)} be the characteristic function of the interval [a, ∞) in ℝ^{+}. For a ≥ 1 let

Ê(z, s) := E(z, s) − χ_{[a,∞)}(y^{s} + c(s)y^{1−s})

be the truncated Eisenstein series. Then Ê(z, s) is square integrable and its norm is given by the Maass-Selberg relation. For r ∈ ℝ, r ≠ 0, we have [Se, (7.42’)]

(2.5)  ∥ Ê(1/2 + ir) ∥^{2}= 2 log a − \frac{c′}{c} (1/2 + ir) + \frac{1}{r} Im (c(1/2 + ir)a^{−2ir}) .

Let ρ = 1/2 + ir, r ≠ 0, be a zero of 1 + c(s)a^{1−2s} of multiplicity ≥ 2. Then

c(1/2 + ir) = −a^{2ir}.

Thus Im (c(1/2 + ir)a^{−2ir}) = 0. This together with (2.4) implies that Ê(1/2 + ir) = 0. On the other hand, the zeroth Fourier coefficient is not identically zero. Hence we arrive at a contradiction.

The case of real zeros is treated similarly. Let σ ∈ (1/2, 1) be a zero of 1 + c(s)a^{1−2s} of multiplicity ≥ 2. In this case the Maass-Selberg relation is

(2.6)  ∥ Ê(σ) ∥^{2}= \frac{a^{2σ−1} − |c(σ)|^{2}a^{1−2σ}}{2σ − 1} + 2c(σ) log a − c′(σ)

[Se, (7.42”)]. Since σ is a zero, we have |c(σ)| = a^{2σ−1}. This together with (2.4) implies Ê(σ) = 0, which again leads to a contradiction.

Finally note that by (2.2) and (2.3), 1/2 is a double zero of φ_{0}(a^{∗}, s). This completes the proof of Theorem 0.1. □

We note that Théorèm 5 of [CV] has an extension to hyperbolic surfaces with several cusps similar to Theorem 1.4. In the case of m cusps the scattering matrix C(s) is an m × m matrix. In this case we study the zeros of the function

(2.7)  ϕ(a, s) := det(a^{s} + a^{1−s}C(s)).

For a congruence subgroup Γ ⊂ PSL(2, ℤ) the entries of C(s) consist of ratios of Dirichlet L-series [Hu]. If we multiply (2.7) by the common denominator, we get a certain linear combination of products of Dirichlet L-functions. As example, consider the case of the Hecke group Γ = Γ_{0}(n), where n = p_{1} ⋯ p_{r} is a product of pairwise distinct prime numbers p_{i}. Then the number of cusps of Γ_{0}(n)\ℍ is m = 2^{r}. The scattering matrix can be described in the following way. For a prime p let

N_{p}(s) := \frac{1}{p^{2s} − 1} (\begin{matrix}p − 1 & p^{s} − p^{1−s} \\ p^{s} − p^{1−s} & p − 1 \end{matrix})

Then

C(s) = \frac{Λ(2s − 1)}{Λ(2s)} (N_{p_{1}}(s) ⊗ ⋯ ⊗ N_{p_{r}}(s))

[He]. Suppose that n = p is a prime number. Then we get

det (a^{s} Id +a^{1−s}C(s))

= a^{2s} + 2a\frac{p − 1}{p^{2s} − 1} · \frac{Λ(2s − 1)}{Λ(2s)} + a^{2−2s} \frac{p^{2−2s} − 1}{p^{2s} − 1} · \frac{Λ(2s − 1)^{2}}{Λ(2s)^{2}}

= \frac{∏_{k=1}^{2} (a^{s}(p^{s} + (−1)^{k})Λ(2s) + a^{1−s}(p^{1−s} + (−1)^{k})Λ(2s − 1))}{(p^{2s} − 1)Λ(2s)^{2}}.

(2.8)

Note that the numerator and the denominator have no common zeros. Therefore the zeros of the function ϕ(a, s) coincide with the zeros of the numerator. Let a ≥ 1. Then it follows as above that the map s ↦ s(1−s) induces a bijection between the zeros ρ ≠ 1/2 of ϕ(a, s) and the eigenvalues μ_{j}(a) ≠ 1/4 of Δ_{a}. This implies that for a ≥ 1 the complex zeros of

a^{s}(p^{s} ± 1)Λ(2s) + a^{1−s}(p^{1−s} ± 1)Λ(2s − 1)

lie all on the line Re(s) = 1/2. As for the real zeros, we observe that Γ_{0}(p) has no residual eigenvalues other than 0 [Iw, Theorem 11.3]. By the mini-max principle it follows that μ_{1}(a) ≥ 1/4. Therefore, there exist at most two real zeros, ρ_{a} and 1 − ρ_{a}, with ρ_{a} ∈ (1/2, 1). Moreover lim_{a→∞} ρ_{a} = 1.

3. Imaginary quadratic fields

In this section we derive Theorem 0.3 from Theorem 1.5.

Let K = ℚ(\sqrt{−D}) be an imaginary quadratic field, where D is a square free positive integer, and let O_{K} be the ring of integers in K. Let Γ_{K} = PSL(2, O_{K}). Then Γ_{K} ⊂ PSL(2, ℂ) is a discrete subgroup of finite co-volume. The number of cusps of Γ_{K}\ℍ^{3} equals the class number h_{K} of K. Assume that h_{K} = 1. This is the case if and only if D = 1, 2, 3, 7, 11, 19, 43, 67, 163 [St]. Then Γ_{K}\ℍ^{3} has a single cusps which we choose to be at κ = ∞. A fundamental domain F_{K} of Γ_{K} can be constructed as in [EGM, Section 7.3], [Sw, pp. 17-19]. Let F_{K} be a fundamental domain for the group

PSL(2, O_{K})_{∞} = \{(\begin{matrix}α & β \\ 0 & λ \end{matrix}) ∈ PSL(2, O_{K})\} .

If μ, λ ∈ O_{K} generate O_{K} and μ ≠ 0, let

H(μ, λ) := {(z, r) ∈ ℍ^{3} : |μz + λ|^{2} + |μ|^{2}r^{2} ≥ 1}.

This is the set of points w ∈ ℍ^{3} which lie above or on the hemisphere S(μ, λ) in ℍ^{3} which is the set of all (z, r) such that |μz + λ|^{2} + |μ|^{2}r^{2} = 1. This hemisphere has center (λ/μ, 0) and radius 1/|μ|. Let

B_{K} := ⋂ H(μ, λ),

where the intersection is taken over all μ, λ ∈ O_{K} as above. Then a fundamental domain F_{K} of Γ_{K} is given by

F_{K} = {(z, r) ∈ B_{K}: z ∈ F_{K}}

[EGM, Theorem 7.3.4], [Sw, Section 3]. Thus F_{K} is the complement in F_{K} × ℝ^{+} of Euclidean semi-balls with center at (λ/μ, 0) and radius 1/|μ|, μ ≠ 0. Since μ ∈ O_{K}, it follows that |μ| ≥ 1 and therefore, we have

(3.1)  F_{K} × [1, ∞) ⊂ F_{K}.

For ℚ(i) and ℚ(\sqrt{−3}), the fundamental domain is a hyperbolic pyramid with one vertex at ∞.

Let d_{K} be the discriminant of K and let ζ_{K}(s) be the Dedekind zeta function of K. Let

Λ_{K}(s) = (\frac{\sqrt{|d_{K}|}}{2π})^{s} Γ(s)ζ_{K}(s)

be the completed zeta function. Then the constant term of the Eisenstein series E(w, s) is given by

(3.2)  E_{0}(y, s) = y^{s} + \frac{Λ_{K}(s − 1)}{Λ_{K}(s)}y^{2−s}

[EGM, Section 8.3], [ES], and therefore, the scattering matrix c_{K}(s) equals

(3.3)  c_{K}(s) = \frac{Λ_{K}(s − 1)}{Λ_{K}(s)}.

The completed zeta function satisfies the functional equation Λ_{K}(s) = Λ_{K}(1 − s). All its zeros are contained in the strip 0 < Re(s) < 1. This implies that Λ_{K}(s − 1) and Λ_{K}(s) have no common zeros. Therefore for a > 0, the zeros of

φ_{K}(a, s) = a^{s}Λ_{K}(s − 1) + a^{2−s}Λ_{K}(s)

coincide with the zeros of E_{0}(a, s).

We can now apply Theorem 1.5. By (3.1) we can take b = 1. Let a ≥ 1 and let ρ ≠ 1 be a zero of ψ_{K}(a, s). Then by Theorem 1.5, 2), ρ(2 − ρ) is an eigenvalue of Δ_{a}. This implies that if ρ(2 − ρ) ≥ 1, then ρ = 1 + ir with r ∈ ℝ, r ≠ 0. If ρ(2 − ρ) < 1, then ρ is real. Thus all complex zeros of φ_{K}(a, s) are on the line Re(s) = 1.

For the real zeros we need to consider the non-cuspidal eigenvalues of Δ. By the spectral resolution of the Laplacian, these eigenvalues are in one-to-one correspondence with the poles of the Eisenstein series E(w, s) in the interval (1, 2] [EGM, Proposition 6.2.2]. On the other hand, using the Maass-Selberg relations, it follows that the poles of E(w, s) in (1, 2] coincide with the poles of the scattering matrix c_{K}(s) in (1, 2]. Since Λ_{K}(s) has no zeros in Re(s) > 1 and the only pole in Re(s) > 0 is a simple pole at s = 1, it follows from (3.3) that the only pole of c_{K}(s) in Re(s) > 1 is a simple pole at s = 2 which corresponds to the eigenvalue 0. This shows that Δ has no non-cuspidal eigenvalues in the interval (0, 1). By Theorem 1.5, 4), it follows that φ_{K}(a, s) has at most two real zeros ρ_{a} and 2−ρ_{a} with ρ_{a} ∈ (1, 2). Let a < a′. Then by Theorem 1.5, 1), we have

−(ρ_{a} − 1)^{2} = ρ_{a}(2 − ρ_{a}) − 1 ≥ ρ_{a′}(2 − ρ_{a′}) − 1 = −(ρ_{a′} − 1)^{2}.

This implies ρ_{a} ≤ ρ_{a′}, and by 4) it follows that lim_{a→∞} ρ_{a} = 1.

Next we consider the eigenvalue 1. Recall that Λ_{K}(s) has simple poles at s = 0 and s = 1. Using the functional equation, we get Res_{s=1}Λ_{K}(s) = −Res_{s=0}Λ_{K}(s). This implies c_{K}(1) = −1. By Theorem 1.5, 3), 1 occurs as an eigenvalue μ_{j}(a) of Δ_{a} if and only if c′_{K}(1) = −2 log a. Let

ξ_{K}(s) = s(s − 1)Λ_{K}(s).

Then

c′_{K}(1) = −\frac{c′_{K}(1)}{c_{K}(1)} = lim_{s→1} (\frac{Λ′_{K}(s)}{Λ_{K}(s)} − \frac{Λ′_{K}(s − 1)}{Λ_{K}(s − 1)}) = −2 (1 + \frac{ξ′_{K}(0)}{ξ_{K}(0)}) .

Let

a_{K}^{∗} := exp (1 + \frac{ξ′_{K}(0)}{ξ_{K}(0)}) .

Then by the above, there is j such that μ_{j}(a) = 1 if and only if a = a_{K}^{∗}. Assume that a_{K}^{∗} ≥ 1. Using that the eigenvalues μ_{j}(a) of Δ_{a} are non-increasing and have multiplicity one, we get μ_{0}(a_{K}^{∗}) = 1, μ_{0}(a) < 1, if a > a_{K}^{∗}, and μ_{0}(a) > 1, if a < a_{K}^{∗}. If a_{K}^{∗} < 1, then μ_{0}(a) < 1 for all 1 ≤ a. Using that μ_{0}(a) = ρ_{a}(2 − ρ_{a}), we get 1) and 2) of Theorem 0.3. The simplicity of the zeros ρ ≠ 1 follows again from the Maass-Selberg relations [EGM, p.270] in the same way as in the case of PSL(2, ℤ). □

The constant a_{K}^{∗} in Theorem 0.3 can be computed as follows. Let w_{K} be the order of the group of units of O_{K}. Let 1, ω be a basis for O_{K} as a ℤ module. Since K has class number one, we have

ζ_{K}(s) = \frac{1}{w_{K}} ∑′_{m,n} \frac{1}{|m + nω|^{2s}} ,

the sum being taken for all integers (m, n) ≠ (0, 0). Using the Kronecker limit formula [La, p. 273], we get for K = ℚ(\sqrt{−D})

(3.4)  a_{K}^{∗} = \frac{e^{2+γ}}{2π\sqrt{D}} \begin{cases}\frac{1}{2|η(i\sqrt{D})|^{4}} , & D ≡ 2, 3 mod 4 \\ _{\frac{1}{|η( \frac{1+i\sqrt{D}}{2} )|^{4}}} , & D ≡ 1 mod 4, \end{cases}

where γ denotes Euler’s constant and η(z) is the Dedekind eta function. The values of the eta function can be computed by the Chowla-Selberg formula [SC, (2), p.110]. Let Δ(z) = η(z)^{24}. Let D ≡ 2, 3 mod 4 and assume that h(D) = 1. Then the Chowla-Selberg formula gives

(3.5)  Δ(\sqrt{D}i) = \frac{1}{(8πD)^{6}} \{∏_{m=1}^{4D} Γ (\frac{m}{4D})^{(^{\frac{−4D}{m}})}\}^{3w_{K}} .

So for D = 1 we get

Δ(i) = \frac{1}{(8π)^{6}} (\frac{Γ (\frac{1}{4})}{Γ (\frac{3}{4})})^{12} = \frac{Γ (\frac{1}{4})^{24}}{2^{6}(2π)^{18}} ,

where in the right equality we used the duplication formula of the Gamma function. This implies

a_{ℚ(i)}^{∗} = \frac{4π^{2}e^{2+γ}}{Γ(1/4)^{4}} ≈ 3.0068.

Similarly for D = 2 we get

Δ(\sqrt{2}i) = \frac{1}{(2π)^{6}8^{6}} (\frac{Γ (\frac{1}{8}) Γ (\frac{3}{8})}{Γ (\frac{5}{8}) Γ (\frac{7}{8})})^{6} = \frac{(Γ (\frac{1}{8}) Γ (\frac{3}{8}))^{12}}{2^{15}(2π)^{18}} ,

and therefore

a_{ℚ(\sqrt{−2})}^{∗} = \frac{8π^{2}e^{2+γ}}{(Γ (\frac{1}{8}) Γ (\frac{3}{8}))^{2}} ≈ 3.2581.

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Universität Bonn, Mathematisches Institut, Beringstrasse 1, D – 53115 Bonn, Germany

  E-mail address: mueller@math.uni-bonn.de