IJPAM: Volume 68, No. 3 (2011)
A CONJECTURE OF HEJHAL
P.O. Box 1424, Princeton, NJ 08542, USA
Abstract. Let . Hejhal has formulated a conjecture for Maass waveforms on , namely, that the ``go Gaussian'' as , where .
Let be a Hecke eigenfunction of
even weight on
In this paper we first replace the with an arbitrary family of independent random variables on an arbitrary probability space and show that within our probability model, if the ``go Gaussian'' as , they do not do so uniformly.
We then briefly discuss a specific model
based on the Sato-Tate probability measure.
Received: November 2, 2010
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