# Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

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## An effective bound for the Huber constant for cofinite Fuchsian groupsHTML articles powered by AMS MathViewer

by J. S. Friedman, J. Jorgenson and J. Kramer
Math. Comp. 80 (2011), 1163-1196 Request permission

## 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: $\pi _M(u) = \sum _{0 \leq \lambda _{M,j} \leq 1/4} \operatorname {li}(u^{s_{M,j}}) + O_M \left (\frac {u^{3/4}}{\log u}\right ),$ 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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• 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
• Email: FriedmanJ@usmma.edu, joshua@math.sunysb.edu, CrownEagle@gmail.com
• 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
• Email: jjorgenson@mindspring.com
• J. Kramer
• Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
• MR Author ID: 227725
• Email: kramer@math.hu-berlin.de
• 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.