Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Conjugacy classes of $\Gamma (2)$ and spectral rigidity
HTML articles powered by AMS MathViewer

by Ralph Phillips PDF
Math. Comp. 64 (1995), 1287-1306 Request permission


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.
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 11F72, 11Y70
  • Retrieve articles in all journals with MSC: 11F72, 11Y70
Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Math. Comp. 64 (1995), 1287-1306
  • MSC: Primary 11F72; Secondary 11Y70
  • DOI:
  • MathSciNet review: 1303088