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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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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: \[ \pi _M(u) = \sum _{0 \leq \lambda _{M,j} \leq 1/4} \operatorname {li}(u^{s_{M,j}}) + O_M \left (\frac {u^{3/4}}{\log u}\right ), \] 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 \[ \left |\pi _{M}(u)-\sum _{0 \leq \lambda _{M,j} \leq 1/4} \operatorname {li}(u^{s_{M,j}})\right | \leq C_{M}\frac {u^{3/4}}{\log {u}} \quad \text {for all $u > 1.$} \] 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
MR Author ID: 772419

J. Jorgenson
Affiliation: Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, New York 10031
MR Author ID: 292611

J. Kramer
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
MR Author ID: 227725

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.