IJPAM: Volume 45, No. 4 (2008)

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

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


Abstract.Determining the $n$-level correlation of the eigenvalues of the hyperbolic Laplacian for the modular surface is a notoriously difficult problem, and very little of substance is known.

In Part I we showed that with regard to one aspect of this problem one gets a dramatically better result by the Bruggeman-Kuznetsov trace formula. In this Part II we show that with regard to another aspect of the problem one gets a dramatically better result by the Selberg trace formula.

In this paper, in addition to the Selberg trace formula, we employ the Rudnick-Sarnak construction, which was created to analyze the $n$-level correlation of the zeros of the Riemann zeta function and principal $L$-functions, together with the weighted Bruggeman-Kuznetsov trace formula, as formulated by Sarnak and Iwaniec.


To the memory of Atle Selberg.


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