ON THE ASYMPTOTIC BEHAVIOR OF COUNTING FUNCTIONS ASSOCIATED TO DEGENERATING HYPERBOLIC RIEMANN SURFACES

摘要

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 greater than or equal to 0 to be [GRAPHICS] where {lambda(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,N-w(T) via the inverse Laplace transform from which one can express the weighted counting function in terms of spectral data associated to M (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. [References: 25]