On the Distribution of the Eigenvalues of the Hyperbolic Laplacian for PSL(2, Z), II Original Research Article

Let H be the upper half plane and X=SL(2, Z)\H the corresponding modular surface. Theory and experiment suggest that the eigenvalues of the hyperbolic Laplacian, Δ, on X, denoted by λj=1/4+t2j, behave in many ways like a random sequence. In particular, for any A>0 the numbers Aλj, j=1, 2, 3, …, should be well distributed modulo 1 (that is to say, there should be square root cancellation in the corresponding Weyl sums). In this paper we show in sharp contrast to the above that the sequence 2tj log(tj/πe) is not well distributed modulo 1. This reflects a certain structure that the closed geodesics on X carry, precisely that the norms of the hyperbolic conjugacy classes (which correspond to closed geodesics) of Γ are very close to being squares of integers. This phenomenon no doubt occurs for all arithmetic quotients X of H but not for the generic hyperbolic surface.