A COMPARISON BETWEEN THE SELBERG AND THE BRUGGEMAN-KUTZNETSOV TRACE FORMULAS III

In this paper we elucidate and make somewhat transparent the clever technique of first introducing and then removing weights (Fourier co- efficients of eigenfunctions) when employing the Bruggeman-Kuznetsov trace formula to obtain information on the distribution of the eigenvalues of the hy- perbolic 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.