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Mathematics of Computation

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

Author: Ralph Phillips
Journal: Math. Comp. 64 (1995), 1287-1306, S35
MSC: Primary 11F72; Secondary 11Y70
MathSciNet review: 1303088
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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

$\displaystyle \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

$\displaystyle \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 $ \vert\theta (x) - x\vert = {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.

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Article copyright: © Copyright 1995 American Mathematical Society