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

AbstractLet 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 2tjlog(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