On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces

In this article we will study what we call weighted counting functions on hyperbolic Riemann surfaces of finite volume. IfMis compact, then we define the weighted counting function forw⩾0 to beNM, w(T)=∑λn⩽T (T−λn)w,where {λn} is the set of eigenvalues of the Laplacian which acts on the space of smooth functions onM. IfMis non-compact, then we define the weighted counting functionNM, w(T) via the inverse Laplace transform from which one can express the weighted counting function in terms of spectral data associated toM(see Proposition 5.1 and Remark 5.2). 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 Mldenote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M0.LetSldenote the set of lengths of pinching geodesics on the family of surfaces Ml.For w⩾0and T⩾1/4,letGl, w(T)=2Γ(w+1)(16π)1/2 ∑n=1∞ ∑lk∈S1 lksinfh(nlk/2) T−1/4(nlk/2)2(w+1/2)/2 Jw+1/2(nlk (T−1/4)),where Js(x)is the J-Bessel function and Γ(x)is the standard gamma function. For0⩽T<1/4,defineGl, w(T)to be zero.Then: (a) Forw>3/2andT>0,we haveNMl, w(T)=Gl, w(T)+NM0, w(T)+o(1) as l→0; (b) For w⩾0andT>0,we haveNMl, w(T)=Gl, w(T)+o ∑lk∈Sl log(1/lk) as l→0; : (c) For w⩾0andT⩾1/4,we haveGl, w(T)=2Γ(w+1)(T−1/4)w+1/2(4π)1/2 Γ(w+3/2) ∑lk∈Sl log(1/lk)+O(1) as l→0.Observe that by takingT<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; see Remark 2.2 and Remark 3.6 for further discussion of the organization of this article.