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An effective bound for the Huber constant for cofinite Fuchsian groups. (English) Zbl 1229.11083

Let \(\Gamma\) be a cofinite Fuchsian group acting on the upper half-plane \(\mathbb H\). For any closed geodesic \(\gamma\) on \(M:=\Gamma\setminus\mathbb H\) let \(\ell(\gamma)\) denote its length and denote by \(\pi(u)\) the number of primitive closed geodesics \(\gamma\) on \(M\) with \(\exp(\ell(\gamma))< u\). Moreover, let \(0= \lambda_0< \lambda_1\leq\cdots\leq\lambda_N\leq\tfrac14\) be the “small” eigenvalues of the hyperbolic Laplacian on \(M\) and put \(s_j:= \tfrac12+\sqrt{\tfrac14- \lambda_j}\) \((j= 0,\dots, N)\). Then the Huber constant for \(\Gamma\) is defined to be the minimal number \(C\) such that \[ \biggl|\pi(u)- \sum^N_{j=0} \ell i(u^{s_j})\biggr|\leq C{u^{3/4}\over\log u} \] for all \(u> 1\). The aim of the paper under review is to estimate \(C\) for an arbitrary cofinite Fuchsian group \(\Gamma\) by means of an effectively computable upper bound involving elementary geometric and spectral theoretic data for \(M\). Since the main result is so long and complicated that it would take two printed pages, the authors present it as an algorithm rather than a formula. As a special case, a numerical bound for \(C\) in the case of the modular group is given.
In the process of proving their main result the authors also determine an explicit constant (depending on \(\Gamma\)) bounding the growth of the so-called spectral counting function of \(\Gamma\). This constant is also given numerically for \(\text{PSL}_2(\mathbb Z)\). Main tools in the proofs are a variant of Karamata’s Tauberian theorem which keeps track of the implicit constants and Selberg’s trace formula.
The results of the present paper complete the analysis of several earlier articles by J. Jorgenson and J. Kramer in providing explicit and effectively computable bounds for various quantities coming up in the Arakelov theory of algebraic curves.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
40E05 Tauberian theorems
40E10 Growth estimates
40E20 Tauberian constants

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