ON THE ASYMPTOTIC BEHAVIOR OF COUNTING

FUNCTIONS ASSOCIATED TO DEGNERATING

HYPERBOLIC RIEMANN SURFACES

Jonathan Huntley  Jay Jorgenson  Rolf Lundelius
Department of Mathematics Department of Mathematics Department of Mathematics
Baruch College CUNY  Yale University  University of the Witwatersrand
17 Lexington Avenue  10 Hillhouse Avenue  Private Bag 3
New York, NY  New Haven, CT 06520  P.O. Wits 2050
USA  USA  Republic of South Africa

December 14, 1994

Abstract. In this article we will study what we call weighted counting functions on hyperbolic Riemann surfaces of finite volume. If M is compact, then we define the weighted counting function for w ≥ 0 to be

N_{M,w}(T) = ∑_{λ_{n}≤T}(T − λ_{n})^{w}

where {λ_{n}} is the set of eigenvalues of the Laplacian which acts on the space of smooth functions on M. If M is non-compact, then we define the weighted counting function N_{M,w}(T) via the inverse Laplace transform. Using the convergence results from [JL2] concerning the regularized heat trace on finite volume hyperbolic Riemann surfaces, we shall prove the following results.

Theorem. Let M_{ℓ} denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M_{0}. For w ≥ 0 and T ≥ 1/4, let

G_{ℓ,w}(T) = \frac{Γ(w + 1)}{(16π)^{1/2}} ∑_{n=1}^{∞} ∑_{ℓ_{k}∈ℓ} \frac{ℓ_{k}}{sinh(nℓ_{k}/2)} (\frac{T − 1/4}{(nℓ_{k}/2)^{2}} )^{(w+1/2)/2} J_{w+1/2}(nℓ_{k}\sqrt{(T − 1/4)})

where J_{s}(x) is the J-Bessel function. For 0 ≤ T < 1/4, define G_{ℓ,w}(T) to be zero. Then:

a) For w > 3/2 and T > 0, we have

N_{M_{ℓ},w}(T) = G_{ℓ,w}(T) + N_{M_{0},w}(T) + o(1) as ℓ → 0;

b) For w ≥ 0 and T > 0, we have

N_{M_{ℓ},w}(T) = G_{ℓ,w}(T) + o (∑ log(1/ℓ_{k})) as ℓ → 0;

Typeset by AMS-TEX

The first author acknowledges support from NSF grant DMS-92-03393, from MSRI under NSF grant DMS-90-22140, and from several PSC-CUNY grants. The second author acknowledges support from NSF grant DMS-93-07023, from the Max-Planck-Institut für Mathematik, and from the Sloan Foundation. Part of the research of the third author was conducted during the US-Sweden Workshop on Spectral Methods sponsored by the NSF under grant INT-9217529.

c) For w ≥ 0 and T ≥ 1/4, we have

G_{ℓ,w}(T) = \frac{Γ(w + 1)(T − 1/4)^{w+1/2}}{(4π)^{1/2}Γ(w + 3/2)} ∑_{k} log(1/ℓ_{k}) + O(1) as ℓ → 0.

Observe that by taking T < 1/4, part (a) proves convergence of certain symmetric functions of the small eigenvalues, thus we prove the main result from [CC]. We discuss a method to improve the error terms in asymptotic expansion of counting functions for 0 ≤ w ≤ 3/2.

§1. Regularized heat traces.

Let M be a connected hyperbolic Riemann surface of finite volume, either compact or non-compact For now, let us assume that M is connected, so then M can be realized as the quotient manifold Γ\h, where h is the hyperbolic plane and Γ is a discrete group of isometries of h. Let K_{h}(t, x̃, ỹ) be the heat kernel on h. We shall assume known that the hyperbolic heat kernel on h is a function of t > 0 and of the hyperbolic distance of x̃ to ỹ, so

K_{h}(t, x̃, ỹ) = K_{h}(t, d_{h}(x̃, ỹ)).

Quoting from page 246 of [Ch], we have, for ρ ≥ 0

K_{h}(t, ρ) = \frac{\sqrt{2}e^{−t/4}}{(4πt)^{3/2}} ∫_{ρ}^{∞} \frac{ue^{−u^{2}/4t}du}{[cosh u − cosh ρ]^{1/2}}

and also

K_{h}(t, 0) = \frac{1}{2π} ∫_{0}^{∞}e^{−(1/4+r^{2})t} tanh(πr)rdr.

The heat kernel K_{M}(t, x, y) on M can be written as a periodization of the heat kernel on the universal cover h, meaning

K_{M}(t, x, y) = ∑_{γ∈Γ} K_{h}(t, x̃, γỹ),

where x̃ and ỹ are any points in h which project via Γ to the points x and y in M. Let H(Γ) denote a set of Γ-inconjugate primitive hyperbolic classes of Γ (meaning classes for which |Tr(γ)| > 2 for any representative γ of the class), and let P (Γ) denote a set of Γ-inconjugate, non-identity, primitive parabolic classes of Γ (meaning classes for which |Tr(γ)| = 2 for any representive γ of the class). If M is compact, then P (Γ) is empty. Let Γ_{γ} denote the centralizer of γ in Γ. Recall that the length ℓ(γ) of the geodesic in the homotopy class determined by γ ∈ Γ satisfies the relation Tr(ℓ(γ)) = 2 sinh(ℓ(γ)/2). Now, we can use elementary group theory, as in the derivation of the Selberg trace formula, to write

K_{M}(t, x, y) = K_{h}(t, x̃, ỹ) + ∑_{n=1}^{∞} ∑_{γ∈P (Γ)} ∑_{κ∈Γ/Γ_{γ}} K_{h}(t, x̃, κ^{−1}γ^{n}κỹ)

+ ∑_{n=1}^{∞} ∑_{γ∈H(Γ)} ∑_{κ∈Γ/Γ_{γ}} K_{h}(t, x̃, κ^{−1}γ^{n}κỹ)

(see [He1], [M], or [S1]). The following theorem defines what we call the hyperbolic heat trace associated to M.

Theorem 1.1. Let M be a connected, hyperbolic Riemann surface of finite volume with p cusps. Let H(Γ) be a set of inconjugate primitive hyperbolic classes of a uniformizing group Γ of M.

a) For each t > 0, the sum

HK_{M}(t, x) = ∑_{n=1}^{∞} ∑_{γ∈H(Γ)} ∑_{κ∈Γ/Γ_{γ}} K_{h}(t, x̃, κ^{−1}γ^{n}κx̃)

is a well-defined function on M.

b) Let ⟨γ⟩ be the cyclic group generated a representative of γ ∈ Γ of a class in H(Γ), and let C_{γ} = ⟨γ⟩\h. Then

HTrK_{M}(t) = ∫_{M} HK_{M}(t, x)dμ(x) = \frac{1}{2} ∑_{γ∈H(Γ)} ∫_{C_{γ}} [K_{C_{γ}} (t, x, x) − K_{h}(t, 0)] dμ(x).

c) Let ℓ(γ) be the length of the geodesic in the homotopy class [γ] determined by γ ∈ Γ. Then

HTrK_{M}(t) = \frac{e^{−t/4}}{(16πt)^{1/2}} ∑_{n=1}^{∞} ∑_{γ∈H(Γ)} \frac{ℓ(γ)}{sinh(nℓ(γ)/2)}e^{−(nℓ(γ))^{2}/4t}.

Outline of Proof. All aspects of Theorem 1.1 follow from work of Selberg [S2] and calculations in McKean [M] and Hejhal [He1]. Since a detailed discussion is given in [JL1], we simply shall outline the main points. Part (a) follows formal group consideration together with elementary growth considerations for the heat kernel on h and for the counting function of geodesics on M (see, in particular, Lemma 1.4 of [JL2]). Part (b) follows from unfolding of the integral and integrand, with the factor of 1/2 appearing since the fundamental group of C_{γ} is isomorphic to Z whereas the corresponding sum in (a) involves only Z^{+} (see also page 233 of [M]). Finally, part (c) follows from the calculations given on page 234 of [M]. □

Definition 1.2. Define the regularized heat trace associated to M by STrK_{M}(t) = HTrK_{M}(t) + vol(M)K_{h}(t, 0).

If M is a hyperbolic Riemann surface of finite volume but not connected, let M_{1}, ⋯, M_{n} be the connected components, and define

HTrK_{M}(t) = ∑_{j=1}^{n} HTrK_{M_{j}}(t) and STrK_{M}(t) = ∑_{j=1}^{n} STrK_{M_{j}}(t).

Observe that if M is compact, then the regularized trace of the heat kernel is simply the trace of the heat kernel.

In [JL3] we gave a construction of a degenerating family M_{ℓ} of either compact or noncompact hyperbolic Riemann surfaces of finite volume. The reader is referred to this article for complete details, which will be assumed here. Given a p-tuple ℓ = (ℓ_{1}, ℓ_{2}, ⋯, ℓ_{p}) of positive real numbers, we say that ℓ → 0 if the sup norm |ℓ| of ℓ approaches zero. Although each M_{ℓ} is connected when ℓ > 0, the limit surface M_{0} need not be connected, and the number of cusps on M_{0} is equal to the number of cusps on M_{ℓ} plus 2p. On a non-connected surface M, if x and y lie on different components, K_{M}(t, x, y) is defined to be zero for all values of t.

Proposition 1.3. Let M_{ℓ} = Γ_{ℓ}\h denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M_{0}. Let DH(Γ_{ℓ}) be the subset of H(Γ_{ℓ}) such that the corresponding geodesic has length approaching zero. Define the degenerating heat trace as the integral

DTrK_{M_{ℓ}}(t) = \frac{1}{2} ∑_{DH(Γ_{ℓ})} ∫_{C_{ℓ_{k}}} [K_{C_{ℓ_{k}}} (x, x, t) − K_{h}(0, t)]dμ(x).

Then for any t > 0 we have the equality

DTrK_{M_{ℓ}}(t) = \frac{e^{−t/4}}{(16πt)^{1/2}} ∑_{n=1}^{∞} ∑_{γ∈DH(Γ_{ℓ})} \frac{ℓ(γ))}{sinh(nℓ(γ)/2)}e^{−(nℓ(γ))^{2}/4t}.

Proof. The result follows directly from Theorem 1.1(c). □

Remark 1.4. Observe that for every primitive, unoriented geodesic on M_{ℓ} with length going to zero, there are two classes in H(Γ_{ℓ}), and the two classes have opposite orientation. Therefore, if we index the sum in Theorem 1.1 by lengths of geodesics rather than elements of H(Γ_{ℓ}), the factor of 1/2 would be eliminated. For another manifestation of this fact, see page 104, both the footnote and line (8.6), of [He1].

The main result of [JL2] is the following convergence theorem.

Theorem 1.5. Let M_{ℓ} denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M_{0}. Let HTrK_{M_{ℓ}}(t) and DTrK_{M_{ℓ}}(t) be the hyperbolic and degenerating heat traces associated to M_{ℓ}.

a) (Pointwise) For fixed t + is with t > 0, we have

lim_{ℓ→0}[STrK_{M_{ℓ}}(t + is) − DTrK_{M_{ℓ}}(t + is)] = STrK_{M_{0}}(t + is).

b) (Uniformity) For fixed t > 0, there is an absolute constant C such that for all s ∈ R the bound

|STrK_{M_{ℓ}}(t + is) − DTrK_{M_{ℓ}}(t + is)| ≤ C(1 + |s|^{3/2})

is uniform in ℓ.

As an immediate application of Theorem 1.5, we have the following bounds.

Theorem 1.6. For any fixed non-compact, finite volume hyperbolic Riemann surface M and fixed t > 0, there is a constant C = C(M) such that

|STrK_{M}(t + is)| ≤ C(1 + |s|)^{3/2}.

Let M_{ℓ} denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M_{0}. Then for any fixed ℓ, there is a constant C = C(ℓ) such that

|DTrK_{M_{ℓ}}(t + is)| ≤ C(ℓ)(1 + |s|)^{3/2}.

Proof. Line (4.2) in [JL2] states the bound

|HTrK_{M}(t + is)| ≤ C(1 + |s|)^{3/2},

from which the first assertion follows since |K_{h}(z, 0)| ≤ K_{h}(t, 0). The second assertion follows from the first and Theorem 1.5(b). □

§2. Asymptotics of spectral measures.

For any function f(t) on R^{+}, we formally define the Laplace transform L(f) and cumulative distribution function F to be

L(f)(z) = ∫_{0}^{∞}e^{−zt}f(t)dt and F (t) = ∫_{0}^{t} f(u)du.

The Laplace transform L(f) of f exists if, for example, f(t) is a piecewise continuous, realvalued function for 0 ≤ t < ∞ and there exists and M and c such that |f(t)| ≤ Me^{ct}. Then L(f)(z) will exist for some complex z in a half-plane Re(z) > a_{0}. Recall that the inverse Laplace transform L^{−1} is

f(u) = \frac{1}{2πi} ∫_{a−i∞}^{a+∞}e^{zt}L(f)(z)dz and F (u) = \frac{1}{2πi} ∫_{a−i∞}^{a+∞}e^{zt}L(f)(z)\frac{dz}{z} .

The inversion formulae holds for any a > a_{0}.

We shall assume that f is such that the Laplace transform L(f) of f exists, and the inversion formula holds. In addition, we shall make the following basic assumption.

Assumption: There is a constant a > 0 such that

∫_{a−i∞}^{a+i∞}(1 + |Im(z)|^{3/2}) |L(f)(z)| \frac{|dz|}{|z|} < ∞.

As a direct application of Theorem 1.5, we have the following result.

Theorem 2.1. Let M_{ℓ} denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M_{0}. Let f be any function satisfying the stated assumption. Let

N_{M_{ℓ},S}(f)(T) = \frac{1}{2πi} ∫_{a−i∞}^{a+i∞}L(f)(z)STrK_{M_{ℓ}}(z)e^{zT} \frac{dz}{z}

and

N_{M_{ℓ},D}(f)(T) = \frac{1}{2πi} ∫_{a−i∞}^{a+i∞}L(f)(z)DTrK_{M_{ℓ}}(z)e^{zT} \frac{dz}{z} .

6 J. HUNTLEY — J. JORGENSON — R. LUNDELIUS

Then

lim_{ℓ→0} [N_{M_{ℓ},S}(f)(T) − N_{M_{ℓ},D}(f)(T)] = N_{M_{0},S}(f)(T).

Proof. The theorem follows from Theorem 1.5 and the dominated convergence theorem. □ §3. Counting functions for w > 3/2.

When considering Theorem 2.1, there is particular interest is the family of functions f_{w}(t) = (w + 1)t^{w}, hence

L(f_{w})(z) = \frac{Γ(w + 2)}{z^{w+1}} and F_{w}(t) = t^{w+1}.

Let

N_{M,w+1}(T) = N_{M,S}((w + 1)t^{w})(T) = \frac{1}{2πi} ∫_{a−i∞}^{a+i∞}\frac{Γ(w + 2)}{z^{w+1}}STrK_{M}(z)e^{zT} \frac{dz}{z} .

We shall call N_{M,w}(T) the w-th counting function of the spectrum of the surface M. By Theorem 1.6, the weighted counting function is defined for w > 3/2. If M is compact, then by Lemma 4.2 below, we can show

N_{M,w}(T) = ∑_{λ_{n}≤T}(T − λ_{n})^{w}.

Directly from Theorem 2.1 above, we have the following result. Theorem 3.1. Let M_{ℓ} denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M_{0}. For any w > 3/2, let

G_{ℓ,w}(T) = N_{M_{ℓ},D}(t^{w−1})(T) = \frac{1}{2πi} ∫_{a−i∞}^{a+i∞}\frac{Γ(w + 1)DTrK_{M}(z)}{z^{w}}e^{zT} \frac{dz}{z} .

Then for T > 0 we have

lim_{ℓ→0} [N_{M_{ℓ},w}(T) − G_{ℓ,w}(T)] = N_{M_{0},w}(T).

Let us now evaluate the asymptotic behavior of the function G_{ℓ,w}(T) for fixed T and w ≥ 0. Theorem 3.2. Let M_{ℓ} denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume.

a) For any w > 3/2 and T ≥ 1/4, we have

G_{ℓ,w}(T) = \frac{Γ(w + 1)}{(16π)^{1/2}} ∑_{n=1}^{∞} ∑_{ℓ_{k}∈ℓ} \frac{ℓ_{k}}{sinh(nℓ_{k}/2)} (\frac{T − 1/4}{(nℓ_{k}/2)^{2}} )^{(w+1/2)/2} J_{w+1/2}(nℓ_{k}\sqrt{(T − 1/4)})

where J_{s}(x) is the J-Bessel function; if T < 1/4, then G_{ℓ,w}(T) = 0. Therefore, G_{ℓ,w}(T) extends to all w ≥ 0.

b) For fixed w ≥ 0 and T ≥ 1/4, we have

G_{ℓ,w}(T) = \frac{Γ(w + 1)(T − 1/4)^{w+1/2}}{(4π)^{1/2}Γ(w + 3/2)} ∑_{k} log(1/ℓ_{k}) + O(1) as ℓ → 0.

Proof. To begin, we need two formulas involving the inverse Laplace transform, namely

L^{−1} [L(f)(s)e^{−bs}] (t) = \{ \begin{matrix}f(t − b) & b ≤ t < ∞ \\ 0 & 0 ≤ t < b \end{matrix}

and, for μ > 0,

L^{−1} [s^{−μ}e^{−k/s}] (t) = (\frac{t}{k} )^{(μ−1)/2} J_{μ−1}(2\sqrt{(kt)}),

where J is the J-Bessel function. There are a number of references for these formulas, one of which is [CRC], line 12 on page 465 and line 80 on page 470. Note that the collapse of the J-Bessel function, namely

J_{−1/2}(x) = \sqrt{(\frac{2}{πx})} cos(x)

allows for further simplification when μ is a half-integer, such as stated in the introduction of [JL2]. In any event, these formulas give the equation

G_{ℓ,w}(T) = \frac{Γ(w + 1)}{(16π)^{1/2}} ∑_{n=1}^{∞} ∑_{ℓ_{k}∈ℓ} \frac{ℓ_{k}}{sinh(nℓ_{k}/2)} (\frac{T − 1/4}{(nℓ_{k}/2)^{2}} )^{(w+1/2)/2} J_{w+1/2}(nℓ_{k}\sqrt{(T − 1/4)})

for T ≥ 1/4 and G_{ℓ,w}(T) = 0 for T < 1/4, as asserted. FFrom this expression, one immediately obtains the continuation of G_{ℓ,w}(T) to any w ≥ 0.

In order to estimate G_{ℓ,w}(T), let us first consider, for each ℓ_{k} ∈ ℓ, the sum

\frac{Γ(w + 1)}{(16π)^{1/2}} ∑_{n≥2/ℓ_{k}} \frac{ℓ_{k}}{sinh(nℓ_{k}/2)} (\frac{T − 1/4}{(nℓ_{k}/2)^{2}} )^{(w+1/2)/2} J_{w+1/2}(nℓ_{k}\sqrt{(T − 1/4)})

Since n ≥ 2/ℓ_{k} and w ≥ 0, we can bound this sum from above by

C_{T} ∑_{n≥2/ℓ} \frac{ℓ_{k}}{sinh(nℓ_{k}/2)}J_{w+3/2}(nℓ_{k}\sqrt{(T − 1/4)}),

for some constant C_{T} which is independent of ℓ_{k}. Now let us use the inequality sinh(x) ≥ e^{x}/4 for x ≥ 1 and the asymptotic formula

J_{p}(x) ∼ \sqrt{(\frac{2}{πx})} cos(x − π/4 − pπ/2),

which we quote from [WW]. Therefore, J_{p}(x) ≤ Cx^{−1/2} ≤ C for x ≥ 1, and the above sum is bounded in absolute value by

C_{T} ∑_{m≥2/ℓ_{k}} ℓe^{−mℓ_{k}/2} ≤ C_{T} ∑_{m=0}^{∞} ℓe^{−mℓ_{k}/2} = \frac{C_{T} ℓ_{k}}{1 − e^{ℓ_{k}/2}} ,

which is bounded independent of ℓ.

It remains to estimate, for each ℓ_{k} ∈ ℓ, the first terms in the series, namely

\frac{Γ(w + 1)}{(16π)^{1/2}} ∑_{n≤2/ℓ_{k}} \frac{ℓ_{k}}{sinh(nℓ_{k}/2)} (\frac{T − 1/4}{(nℓ_{k}/2)^{2}} )^{(w+1/2)/2} J_{w+1/2}(nℓ_{k}\sqrt{(T − 1/4)}).

For this, we need the estimates

\frac{1}{sinh x} = \frac{1}{x} + O(1/x) and J_{p}(x) = \frac{x^{p}}{Γ(p + 1)2^{p}} + O(x^{p+2}) as x → 0.

Putting these estimates together, we have

G_{ℓ,w}(T) = \frac{Γ(w + 1)(T − 1/4)^{w+1/2}}{(16π)^{1/2}Γ(w + 1/2)} ∑_{k} ∑_{0≤n≤2/ℓ_{k}} (\frac{2}{n} + O(nℓ_{k}^{2})) (1 + O((nℓ_{k})^{2})) + O(1)

= \frac{Γ(w + 1)(T − 1/4)^{w+1/2}}{(4π)^{1/2}Γ(w + 1/2)} ∑_{k} log(1/ℓ_{k}) + O(1),

which completes the proof of the theorem. □

One very interesting aspect of Theorem 3.2 is that G_{ℓ,w}(T) is zero for T < 1/4. Therefore, Theorem 3.2 admits the following corollary.

Corollary 3.3. Let M_{ℓ} denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M_{0}. For any T < 1/4 and w > 3/2, we have

lim_{ℓ→0} N_{M_{ℓ},w}(T) = N_{M_{0},w}(T).

Remark 3.4. From Proposition 4.1 and Proposition 5.1 below, one can write, for any T < 1/4,

N_{M,w}(T) = ∑_{λ_{n}≤T}(T − λ_{n})^{w},

which means that Corollary 3.3 asserts convergence of certain symmetric functions of the small eigenvalues, which are eigenvalues less than 1/4. This result has been studied elsewhere, namely [CC] and [He1].

Remark 3.5. Let us define the function

c_{w}(T) = \frac{Γ(w + 1)(T − 1/4)^{w+1/2}}{(4π)^{1/2}Γ(w + 3/2)} .

Using the identity sΓ(s) = Γ(s + 1), we have

\frac{d}{dT} c_{w+1}(T) = (w + 1)c_{w}(T).

Observe that for w ≥ 0, the function c_{w}(T) is monotone increasing in T. These observations will be important in the following section.

§4. Counting functions for 0 ≤ w ≤ 3/2: compact case

In the previous section, we obtained asymptotic behavior of the weighted counting functions for w > 3/2. In this section, we consider weighted counted functions for 0 ≤ w ≤ 3/2. To begin, we need to improve the bounds given in Theorem 1.6, which is accomplished by using the spectral interpretation of the regularized heat trace as given to us from the Selberg trace formula. FFrom page 44 of [He2], we have the following result.

Proposition 4.1. If M is a compact surface, then the regularized heat trace has the spectral realization

STrK_{M}(t) = ∑_{λ_{n}} e^{−λ_{n}t}.

From Proposition 4.1, we have that the inverse Laplace transforms which define the weighted counting functions converge for all w > 0. Since

\frac{d}{dT} N_{M,w+1}(T) = (w + 1)N_{M,w}(T)

for any w > 0, we define N_{M,0}(T) via this differential equation. It is elementary to show

N_{M,0}(T) = ∑_{λ_{n}≤T} 1,

where the eigenvalues are counted with multiplicities.

Lemma 4.2. For any function f satisfying the assumptions of Theorem 2.1, we have

∫_{0}^{T} F (T − u)L^{−1} (STrK) (u)du = ∑_{0≤λ_{n}≤T} F (T − λ_{n}).

Therefore, the distribution L^{−1} (STrK) (u) is positive, by which we mean if f is a positive function, the above integral is positive and increasing in T.

Proof. Directly from Theorem 8.1 (page 73) and Theorem 12.1(a) (page 91) of [Wi], we have

N_{S,M}(f)(T) = \frac{1}{2πi} ∫_{a−i∞}^{a+i∞}L(f)(z)STr_{M}K(z)e^{T z} \frac{dz}{z} = ∫_{0}^{T} F (T − u)L^{−1} (STrK(t)) (u)du

and, by Lemma 4.2 and the definition of the Laplace transform,

∫_{0}^{T} F (T − u)L^{−1} (STrK(t)) (u)du = ∑_{0≤λ_{n}≤T} F (T − λ_{n}),

as asserted. Since f is positive, F is positive and increasing, from which the result follows. □

Lemma 4.3. For any T > 0 and w ≥ 0, we have

N_{w}(T) ≤ \frac{1}{w + 1} \frac{N_{w+1}(T + ϵ) − N_{w+1}(T)}{ϵ} ≤ N_{w}(T + ϵ).

Proof. By Lemma 4.2, the function N_{M,w}(T) is a monotone for any w ≥ 0. With this, both inequalities follow from the mean value theorem. □

The main result of this section is the following theorem.

Theorem 4.4. Let M_{ℓ} denote a degenerating family of compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M_{0}. Then for any w ≥ 0, and T ≥ 1/4, we have

N_{w}(T) ∼ c_{w}(T) ∑ log(1/ℓ_{k}).

Proof. For fixed w > 1/2, T ≥ 1/4, and ℓ > 0, consider the function

h_{w,ℓ}(T) = \frac{N_{w}(T)}{∑ log(1/ℓ_{k})}.

From the inequality in Lemma 4.3, we have

\frac{1}{w + 1} \frac{h_{w+1}(T + ϵ) − h_{h+1}(T)}{ϵ} ≤ h_{w,ℓ}(T + ϵ).

Now let ℓ → 0 and use Theorem 3.1 and Theorem 3.2 to get

\frac{1}{w + 1} \frac{c_{w+1}(T + ϵ) − c_{w+1}(T)}{ϵ} ≤ lim inf_{ℓ→0} h_{w,ℓ}(T + ϵ).

The right hand side is a decreasing function in ϵ, hence is continuous almost everywhere. Therefore, if we let ϵ → 0, we get the inequality

\frac{1}{w + 1} \frac{dc_{w+1}}{dT} (T) ≤ lim inf_{ℓ→0} h_{w,ℓ}(T)

for almost all T. Similarly, we have, for almost all T

lim sup_{ℓ→0} h_{w,ℓ}(T) ≤ \frac{1}{w + 1} \frac{dc_{w+1}}{dT} (T).

Upon combining, we get the inequality

lim sup_{ℓ→0} h_{w,ℓ}^{+}(T) ≤ \frac{1}{w + 1} \frac{dc_{w+1}}{dT} (T) ≤ lim inf_{ℓ→0} h_{w,ℓ}(T)

which holds for almost all T. Since the reverse inequalities hold by definition, we have, for almost all T,

lim_{ℓ→0} h_{w,ℓ}(T) = \frac{1}{w + 1} \frac{dc_{w+1}}{dT} (T) = c_{w}(T)

As observed in Remark 3.5, the function c_{w}(T) is continuous and monotone, hence the above equality necessarily holds for all T.

We now have established the validity of theorem for all w > 1/2. We can now repeat the argument for any w ≥ 0 to finish the proof. □

Remark 4.5. For other values of w ≤ 3/2, the above analysis gives no estimate concerning the error term in the asymptotic expansion of the weighted counting function, unlike the work in [JiZ]. For related remarks, see section 6 below. However, as we shall see in the next section, the above analysis applies to degenerating non-compact hyperbolic Riemann surfaces, a setting not covered by the methods of proof in [He1], [JiZ], or [Wo].

When considering T < 1/4 as in Corollary 3.3, the proof of Theorem 4.4 does apply to prove the following result.

Corollary 4.6. Let M_{ℓ} denote a degenerating family of compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M_{0}. If w > 0 and T < 1/4 , then we have the limit

lim_{ℓ→0} N_{M_{ℓ},w}(T) = N_{M_{0},w}(T).

If T < 1/4 is not an eigenvalue of the Laplacian on the limit surface, then we have the limit

lim_{ℓ→0} N_{M_{ℓ},0}(T) = N_{M_{0},0}(T).

With Corollary 4.6, we obtain many of the results of [CC].

§5. Counting functions for 0 ≤ w ≤ 3/2: non-compact case

The aim of this section is to prove Theorem 4.4 in the case M_{ℓ} is a degenerating family of non-compact surfaces. The method of proof is similar to that of section 4. That is, we shall first quote the Selberg trace formula to get a spectral interpretation of the regularized heat trace. This will allow us to define the spectral counting function for a non-compact surface whenever w ≥ 0. After this, we shall prove a positivity result associated to the regularized heat trace. After this, the argument of Theorem 4.4 will apply in the non-compact case.

From page 313 of [He3], we have the following spectral interpretation of the regularized heat trace.

Proposition 5.1. If M is a non-compact surface with p > 0 cusps, then the regularized heat trace has the spectral realization

STrK_{M}(t) = ∑_{C(M)} e^{−λ_{n}t} − \frac{1}{4π} ∫_{−∞}^{∞} e^{−(r^{2}+1/4)t}ϕ′/ϕ(1/2 + ir)dr

+ \frac{p}{2π} ∫_{−∞}^{∞} e^{−(r^{2}+1/4)t}Γ′/Γ(1 + ir)dr − \frac{1}{4}(p − TrΦ(1/2))e^{−t/4} + \frac{p log 2}{\sqrt{4πt}}e^{−t/4},

where C(M) denotes the (possibly finite) set of eigenvalues corresponding to L^{2} eigenfunctions on M, and ϕ(s) is the determinant of the scattering matrix Φ(s)

From Proposition 5.1, we have that the inverse Laplace transforms which define the weighted counting functions converge for all w > 0. From the differential equation

\frac{d}{dT} N_{M,w+1}(T) = (w + 1)N_{M,w}(T),

we can define the weighted counting function for all w ≥ 0. It is an easy exercise, which we leave, to express the counting function N_{M,0}(T) as we define in terms of the spectral data given in Proposition 5.1. We omit the exercise because, as we shall see in the proof of Theorem 5.4, the only factors from Proposition 5.1 which significantly contribute to N_{M,w}(T) are those factors involving the eigenvalues C(M) and the scattering determinant ϕ.

From pages 72-74 and 160-161 of [He3], specifically line (12.4) and Proposition 12.7, we have the following result.

Lemma 5.2. Let {s_{k}} be the set of numbers for which 1/2 ≤ s_{k} ≤ 1 and s_{k}(1 − s_{k}) = λ_{k} ≤ 1/4 is a non-cuspidal eigenvalue of the Laplacian on M. Then there is a constant q_{M} ≥ 0 such that for all r,

−ϕ′/ϕ(1/2 + ir) − ∑_{k=1}^{N} \frac{1 − 2s_{k}}{(s_{k} − 1/2)^{2} + r^{2}} − 2 log q_{M} ≥ 0.

Remark 5.3. As on page 160 of [He3], the constant q_{M} can be described as follows. The scattering determinant can be written as

ϕ(s) = (\frac{π^{−(s−1/2)}Γ(s − 1/2)}{π^{−s}Γ(s)})^{p} L(s) where L(s) = ∑_{q} \frac{c(q)}{q^{s}}

for some set of positive numbers {q} tending to infinity. Then by combining the lines (12.3) on page 160, (12.8) on page 166, and the proof and Proposition 12.7 on page 161, we have

log q_{M} = inf_{q} log q.

As remarked on page 16 of [S2], log q_{M} ≤ 0. Of course, once we have the validity of Lemma 5.2 for one value of q_{M}, the inequality holds when replacing q_{M} by any smaller, positive number. As we shall see, it is necessary for us to assume that log q_{M} ≤ 0.

We now can argue as in the compact setting of the previous section and obtain the lead term asymptotics of the spectral counting function for degenerating families of non-compact hyperbolic Riemann surfaces.

Theorem 5.4. Let M_{ℓ} denote a degenerating family of non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M_{0}. Then for any w ≥ 0, and T ≥ 1/4, we have

N_{M_{ℓ},w}(T) ∼ c_{w}(T) ∑ log(1/ℓ_{k}).

Equivalently, we have the following statement. For any non-compact surface M, let

ŜTrK_{M}(t) = ∑_{C(M)} e^{−λ_{n}t} − \frac{1}{4π} ∫_{−∞}^{∞} e^{−(r^{2}+1/4)t}ϕ′/ϕ(1/2 + ir)dr.

Define the distribution

ν_{M}(u) = L^{−1}(ŜTrK_{M})(u)

and, for w ≥ 0, set

{ \atop N_{M,w}}(T) = ∫_{0}^{T}(T − u)^{w}ν_{M}(u)du.

Then

{ \atop N_{M_{ℓ},w}}(T) ∼ c_{w}(T) ∑ log(1/ℓ_{k}).

Proof. In the notation of Proposition 5.1 and Lemma 5.2, let us write the regularized heat trace for any non-compact surface M as

STrK_{M}(t) = g_{M,1}(t) + g_{M,2}(t) + g_{M,3}(t)

where:

g_{M,1}(t) = ∑_{C(M)} e^{−λ_{n}t} − \frac{1}{4π} ∫_{−∞}^{∞} e^{−(r^{2}+1/4)t}ϕ′/ϕ(1/2 + ir)dr

− ∑_{k=1}^{N} ∫_{−∞}^{∞} e^{−(r^{2}+1/4)t} \frac{(1 − 2s_{k})}{4π[(s_{k} − 1/2)^{2} + r^{2}]}dr − \frac{log q_{M}}{\sqrt{4πt}} e^{−t/4};

g_{M,2}(t) = ∑_{k=1}^{N} ∫_{−∞}^{∞} e^{−(r^{2}+1/4)t} \frac{(1 − 2s_{k})dr}{4π[(s_{k} − 1/2)^{2} + r^{2}]} + \frac{log q_{M}}{\sqrt{4πt}} e^{−t/4};

g_{M,3}(t) = \frac{p}{2π} ∫_{−∞}^{∞} e^{−(r^{2}+1/4)t}Γ′/Γ(1 + ir)dr − \frac{1}{4}(p − TrΦ(1/2))e^{−t/4} + \frac{p log 2}{\sqrt{4πt}}e^{−t/4}.

With this notation, we can write the counting function N_{M_{ℓ},w}(T) as

N_{M_{ℓ},w}(T) = ∫_{0}^{T}(T − u)^{w} [L^{−1}(g_{M,1})(u) + L^{−1}(g_{M,2})(u) + L^{−1}(g_{M,3})(u)] du,

as well as

{ \atop N_{M_{ℓ},w}}(T) = ∫_{0}^{T}(T − u)^{w} [L^{−1}(g_{M,1})(u) + L^{−1}(g_{M,2})(u)] du,

By Lemma 5.2, the function

I_{M,1}(T; w) = ∫_{0}^{T}(T − u)^{w}L^{−1}(g_{M,1})(u)du

is monotone increasing. Since log q_{M} ≤ 0 and each s_{k} ∈ [1/2, 1], the function

I_{M,2}(T; w) = ∫_{0}^{T}(T − u)^{w}L^{−1}(g_{M,2})(u)du

is monotone decreasing.

Assume w > 1/2, T > 1/4 is fixed, and let ϵ > 0. By the mean value theorem and the above monotonicity statements, there exist constants ξ_{1}, ξ_{2} ∈ [T, T + ϵ] such that

I_{M_{ℓ},1}(T; w) + I_{M_{ℓ},2}(T + ϵ; w) + I_{M_{ℓ},3}(ξ_{1}; w) ≤ \frac{N_{M_{ℓ},w+1}(T + ϵ) − N_{M_{ℓ},w+1}(T)}{ϵ(w + 1)}

≤ I_{M_{ℓ},1}(T + ϵ; w) + I_{M_{ℓ},2}(T; w) + I_{M_{ℓ},3}(ξ_{2}; w).

Since |Tr(Φ(1/2))| ≤ p, the function I_{M_{ℓ},3}(ξ; w) can be bounded independently of ℓ. Therefore, we have the limit

lim_{ℓ→0} \frac{I_{M_{ℓ},3}(ξ; w)}{∑ log(1/ℓ_{k})} = 0,

hence, we immediately have the limit

lim_{ℓ→0} (\frac{N_{M_{ℓ,w}}(T)}{∑ log(1/ℓ_{k})} − \frac{{ \atop N_{M_{ℓ},w}}(T)}{∑ log(1/ℓ_{k})}) = 0,

which proves the equivalence asserted in the statement of the theorem. With this, we can use Theorem 3.1 and Theorem 3.2 to obtain the inequalities

lim sup_{ℓ→0} (\frac{I_{M_{ℓ},1}(T; w)}{∑ log(1/ℓ_{k})} + \frac{I_{M_{ℓ},2}(T + ϵ; w)}{∑ log(1/ℓ_{k})}) ≤ \frac{1}{w + 1} \frac{c_{w+1}(T + ϵ) − c_{w+1}(T)}{ϵ}

≤ lim inf_{ℓ→0} (\frac{I_{M_{ℓ},1}(T + ϵ; w)}{∑ log(1/ℓ_{k})} + \frac{I_{M_{ℓ},2}(T; w)}{∑ log(1/ℓ_{k})}) .

If ϵ → 0 with ϵ > 0, then the lower bound is monotone increasing, and the upper bound is monotone decreasing, since the bounds are limits of monotone functions. Therefore, the upper and lower bounds are continuous almost everywhere; that is, for almost all T, we have

lim sup_{ℓ→0} (\frac{I_{M_{ℓ},1}(T; w)}{∑ log(1/ℓ_{k})} + \frac{I_{M_{ℓ},2}(T; w)}{∑ log(1/ℓ_{k})}) ≤ \frac{1}{w + 1} \frac{dc_{w+1}}{dT} (T)

≤ lim inf_{ℓ→0} (\frac{I_{M_{ℓ},1}(T; w)}{∑ log(1/ℓ_{k})} + \frac{I_{M_{ℓ},2}(T; w)}{∑ log(1/ℓ_{k})}) .

Since

{ \atop N_{M_{ℓ},w}}(T) = I_{M_{ℓ},1}(T; w) + I_{M_{ℓ},2}(T; w),

we have shown

lim sup_{ℓ→0} (\frac{{ \atop N_{M_{ℓ},w}}(T)}{∑ log(1/ℓ_{k})}) ≤ \frac{1}{w + 1} \frac{dc_{w+1}}{dT} (T) ≤ lim inf_{ℓ→0} (\frac{{ \atop N_{M_{ℓ},w}}(T)}{∑ log(1/ℓ_{k})}) .

The reverse inequalities are trivial. Therefore, we have, for almost all T, the existence of the following limit, and the equality

lim_{ℓ→0} (\frac{{ \atop N_{M_{ℓ},w}}(T)}{∑ log(1/ℓ_{k})}) = \frac{1}{w + 1} \frac{dc_{w+1}}{dT} (T) = c_{w}(T).

Finally, because the function c_{w}(T) is continuous and monotone increasing in T, the equality necessarily holds for all T.

Having proved the stated theorem for w > 1/2, we can repeat the argument for any w ≥ 0, which completes the proof of the theorem. □

Remark 5.5. By Theorem 5.4, the analogue of Corollary 3.3 and Corollary 4.6 hold in the non-compact setting as well.

§6. Improved error terms for 0 ≤ w ≤ 3/2.

It is seems plausible that one could improve the error term in the asymptotic behavior of the counting functions N_{M_{ℓ},w}(T) for w ≤ 3/2 by using arguments as above together with the complication that ϵ approaches zero at a rate which depends on ℓ. However, it appears that such results would require one to improve the implied error term in Theorem 1.5(a), which would in turn improve the implied error term in Theorem 3.1. Let us briefly discuss how such results can be obtained.

Assume that for w > 1/2 and T > 1/4, one has the expansion

N_{M_{ℓ},w+1}(T) = G_{ℓ,w+1}(T) + N_{M_{0},w+1}(T) + O(f(ℓ)) as ℓ → 0,

where f(ℓ) is a function which approaches zero as ℓ approaches zero. We can argue as in section 4 to get the upper bound

(w + 1)N_{M_{ℓ},w}(T) ≤ \frac{N_{M_{0},w+1}(T + ϵ) − N_{M_{0},w+1}(T)}{ϵ}

+ \frac{G_{ℓ,w+1}(T + ϵ) − G_{ℓ,w+1}(T)}{ϵ} + O (f(ℓ)/ϵ(ℓ)) .

By the mean value theorem, we can write

N_{M_{ℓ},w}(T) ≤ N_{M_{0},w}(T) + ϵ(ℓ)\frac{d}{dT} N_{M_{0},w}(ξ_{1}) + G_{ℓ,w}(T) + ϵ(ℓ)\frac{d}{dT} G_{ℓ,w}(ξ_{2}) + O (f(ℓ)/ϵ(ℓ))

for some ξ_{1}, ξ_{2} ∈ [T, T + ϵ]. By similar computations, we get

N_{M_{ℓ},w}(T) ≥ N_{M_{0},w}(T) + ϵ(ℓ)\frac{d}{dT} N_{M_{0},w}(ξ_{3}) + G_{ℓ,w}(T) + ϵ(ℓ)\frac{d}{dT} G_{ℓ,w}(ξ_{4}) + O (f(ℓ)/ϵ(ℓ))

for some ξ_{3}, ξ_{4} ∈ [T − ϵ, T]. Theorem 3.2 now applies. In the end, we have the asymptotic formula

N_{M_{ℓ},w}(T) = N_{M_{0},w}(T) + G_{ℓ,w}(T) + O (ϵ(ℓ) ∑ log(1/ℓ_{k})) + O (f(ℓ)/ϵ(ℓ)) .

We seek to let ϵ = ϵ(ℓ) approach zero so that the maximum of the above error terms is minimized. Inductively, one then is able to successively improve the errors for w > 1/2 and then for w ≥ 0. Detailed investigation into the error term in Theorem 1.5(a) will be considered elsewhere.

Remark 6.1. In this paper, we considered degenerating hyperbolic Riemann surfaces, so the groups under consideration did not contain elliptic elements. By similar methods, we can address the problem of counting functions associated to the spectral theory of degenerating discrete groups with a fixed set of elliptic elements. In a related question, the spectral asymptotics for degenerating groups with elliptic elements of increasing order, analogous to the results in [JL2], will be presented elsewhere.

Acknowledgements. The second author (J. J.) encountered weighted counting functions in collaboration with Serge Lang, based on a course taught by Lang in the academic year 1991-1992 at Yale. At the time, we introduced a technique from analytic number theory, using these counting functions, into our more general context applying also to spectral theory. For instance, we transferred to the general case Theorem 30 on page 82 of [I] and the technique used there. In the present paper, the authors use similar higher weighted counting functions combined with the inverse Laplace transform stemming from work in [W]. The second author very gratefully acknowledges many helpful conversations with Serge Lang concerning the work in the present article and other work. Also, the authors thank Werner Müller whose careful and critical proofreading of [JL2] was one of the factors leading to the considerations in the present article.

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