Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Conjugacy classes of $\Gamma (2)$ and spectral rigidity


Author: Ralph Phillips
Journal: Math. Comp. 64 (1995), 1287-1306
MSC: Primary 11F72; Secondary 11Y70
DOI: https://doi.org/10.1090/S0025-5718-1995-1303088-5
MathSciNet review: 1303088
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The free group $\Gamma (2)$ is generated by $A = (1\;2,0\;1)$ and $B = (1\;0, - 2\;1)$, and setting ${\chi _{(\xi ,\eta )}}(A) = \exp (2\pi i\xi ),{\chi _{(\xi ,\eta )}}(B) = \exp (2\pi i\eta )$ defines a unitary character on $\Gamma (2)$ for $0 \leq \xi ,\eta < 1$. A program is devised to compute \[ \mu ({\text {tr}}) = \sum {{\chi _{(\xi ,\eta )}}({\text {conj}}. {\text {class}}),} \] summed over all primitive conjugacy classes of $\Gamma (2)$ of trace tr. Combined with a Luo-Sarnak theorem, this yields lower bounds for the spectral variance for a large sampling of characters in the range $0 < \xi ,\eta < 1$. The results indicate that the Berry conjecture for spectral rigidity does not hold for this set of classically chaotic systems. The program is also used to compute \[ \theta (x) = \sum {\ln (N(\{ \gamma \} )),} \] summed over all primitive conjugacy classes of $\Gamma (2)$ of norm $N(\{ \gamma \} ) \leq x$. The function $\theta (x)$ is asymptotic to x, and the remainder can be written as $|\theta (x) - x| = {x^\beta }$. The values of $\beta (x)$ are computed for all traces between 3202 and 4802 (here $x = {\text {tr}^2} - 2$). The $\beta$’s cluster around 0.6, attaining a maximum of $2/3$. Finally, it is proved that the remainder $\theta (x) - x$ has a negative bias by showing that the mean normalized remainder converges to a negative limit.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11F72, 11Y70

Retrieve articles in all journals with MSC: 11F72, 11Y70


Additional Information

Article copyright: © Copyright 1995 American Mathematical Society