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An effective bound for the Huber constant for cofinite Fuchsian groups

Authors: J. S. Friedman, J. Jorgenson and J. Kramer
Journal: Math. Comp. 80 (2011), 1163-1196
MSC (2010): Primary 11F72; Secondary 30F35
Published electronically: October 28, 2010
MathSciNet review: 2772118
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Abstract: Let $ \Gamma$ be a cofinite Fuchsian group acting on hyperbolic two-space $ \mathbb{H}$. Let $ M=\Gamma \setminus \mathbb{H}$ be the corresponding quotient space. For $ \gamma,$ a closed geodesic of $ M$, let $ l(\gamma)$ denote its length. The prime geodesic counting function $ \pi_{M}(u)$ is defined as the number of $ \Gamma$-inconjugate, primitive, closed geodesics $ \gamma $ such that $ e^{l(\gamma)} \leq u.$ The prime geodesic theorem states that:

$\displaystyle \pi_{M}(u)=\sum_{0 \leq \lambda_{M,j} \leq 1/4}$   li$\displaystyle (u^{s_{M,j}}) + O_{M}\left(\frac{u^{3/4}}{\log{u}}\right), $

where $ 0=\lambda_{M,0} < \lambda_{M,1} < \cdots $ are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth functions on $ M$ and $ s_{M,j} = \frac{1}{2}+\sqrt{\frac{1}{4} - \lambda_{M,j} }$. Let $ C_{M}$ be the smallest implied constant so that

$\displaystyle \left\vert\pi_{M}(u)-\sum_{0 \leq \lambda_{M,j} \leq 1/4} \text{l...\vert \leq C_{M}\frac{u^{3/4}}{\log{u}} \quad \text{\text{for all} $u > 1.$ }$

We call the (absolute) constant $ C_{M}$ the Huber constant.

The objective of this paper is to give an effectively computable upper bound of $ C_{M}$ for an arbitrary cofinite Fuchsian group. As a corollary we bound the Huber constant for $ \PSL(2,\mathbb{Z}),$ showing that $ C_{M} \leq 16,607,349,020,658 \approx \exp(30.44086643)$.

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Additional Information

J. S. Friedman
Affiliation: Department of Mathematics and Sciences, United States Merchant Marine Academy, 300 Steamboat Road, Kings Point, New York 11024

J. Jorgenson
Affiliation: Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, New York 10031

J. Kramer
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany

Received by editor(s): September 25, 2009
Received by editor(s) in revised form: March 2, 2010
Published electronically: October 28, 2010
Additional Notes: The second named author acknowledges support from grants from the NSF and PSC-CUNY.
The third named author acknowledges support from the DFG Graduate School Berlin Mathematical School and from the DFG Research Training Group Arithmetic and Geometry.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.