On the pair correlation of the eigenvalues of the hyperbolic Laplacian for \(\text{PSL}(2,\mathbb Z)\setminus \mathbb{H}\). (Q1421299)

Denote by \(\lambda_j={1\over 4}+ t^2_j\) (\(j\geq 1\); \(t_j> 0\)) the non-trivial eigenvalues of the Laplacian on the modular surface \(X= \text{PSL}_2(\mathbb{Z})\setminus\mathbb{H}\). The author evaluates the function \[ E(\alpha, T)= \sum^\infty_{j=1} e^{-t^2_j/T^2}\cos(\alpha Tt_j) \] (\(\alpha> 0\), \(T\geq 2\)) by means of the Selberg trace formula. The result does not seem to be useful for the investigation of the asymptotics of \(E(\alpha,T)\) for \(T\to\infty\) since the ``error term'' is of the same order of magnitude as the leading term. Hence the present approach to the pair correlation problem does not yield a result of the desired type.