On the pair correlation of the eigenvalues of the hyperbolic Laplacian for [formula omitted

Abstract

Let H be the upper half plane and X=PSL(2, Z )⧹H the corresponding modular surface. The eigenvalues of the hyperbolic Laplacian, Δ, on X are denoted by λ j =1/4+ t j 2. For α>0 and T⩾2 let E(α,T)= ∑ 0<t j e − t j 2 T 2 cos(αTt j). In this paper, we evaluate E( α, T) by means of the Selberg trace formula. Since the contribution to the Selberg trace formula from the hyperbolic conjugacy classes is not manageable, we have the result that the Selberg trace formula, the seemingly natural tool to use to attack the pair correlation problem, does not yield meaningful results when used with a natural choice of test function. In combining the results in this paper with the results in Part II, substantial evidence is presented to indicate that the Selberg trace formula may not be the natural tool to use to attack the problem, after all.