Abstract.In this paper we elucidate and make somewhat transparent the clever technique of first introducing and then removing weights (Fourier coefficients of eigenfunctions) when employing the Bruggeman-Kuznetsov trace formula to obtain information on the distribution of the eigenvalues of the hyperbolic Laplacian for the modular group.

Frequently, this technique yields improvement of results obtained by the Selberg trace formula. This gain is realized because the sums on the geometric side of the Bruggeman-Kuznetsov trace formula involve sums and integrals, which apparently package certain cancellations in a more efficient way than do the sums involving class numbers, which appear naturally on the geometric side of the Selberg trace formula.

We do this by elaborating and significantly modifying the argument outlined in a letter from Sarnak to Rudnick, and, in the process, we improve one of the results obtained there. The limit of the construction is also discussed.

To the memory of my friend Paul Cohen.

Received: April 10, 2009

AMS Subject Classification: 11F03, 11F11, 11F12, 11F72

Key Words and Phrases: modular group, pair correlation, eigenvalues, Laplacian, Selberg trace formula, Bruggeman-Kuznetsov trace formula

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 53
Issue: 2