IJPAM: Volume 45, No. 4 (2008)
THE BRUGGEMAN-KUZNETSOV TRACE FORMULAS
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 goes to
infinity for two functions
and
for some
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
and
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