IJPAM: Volume 45, No. 4 (2008)

A COMPARISON BETWEEN THE SELBERG AND
THE BRUGGEMAN-KUZNETSOV TRACE FORMULAS

C.J. Mozzochi
P.O. Box 1424, Princeton, NJ 08542, USA
e-mail: cjm@ix.netcom.com


Abstract.In this paper we further 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 obtaining meaningful expansions as $T$ goes to infinity for two functions $E^*(\alpha,T)$ and $F^*(\alpha,T)$ for some $\alpha\in\dbR$ by means of the Bruggeman-Kuznetsov trace formula. In two previous papers we have shown that one is not able to obtain a meaningful expansion for corresponding functions $E(\alpha,T)$ and $F(\alpha,T)$ by means of the Selberg trace formula because of the limitations in the presently available technique for estimating the hyperbolic classes contribution to said formula.

Received: April 21, 2008

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: 2008
Volume: 45
Issue: 4