= Selberg trace formula
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Let $X=\Gamma\backslash\mathbb H^2$ be a <closed> orientable <hyperbolic surface>, where $\Gamma\leq PSL_2(\mathbb R)$ is a <torsion-free> <discrete subgroup>. Write $\lambda_j$ for the nonnegative <eigenvalues> of its <Laplace-Beltrami operator>, with multiplicities. A heat version of the Selberg trace formula is, for $t>0$,
$$
\sum_{j=0}^{\infty}e^{-t\lambda_j}
=\frac{\operatorname{area}(X)}{4\pi}\int_{-\infty}^{\infty}r\tanh(\pi r)e^{-t(r^2+1/4)}\,dr
+\frac{e^{-t/4}}{\sqrt{4\pi t}}\sum_{[\gamma]\ \mathrm{primitive}}\sum_{k=1}^{\infty}\frac{\ell_\gamma\,e^{-(k\ell_\gamma)^2/(4t)}}{2\sinh(k\ell_\gamma/2)}.
$$
The outer geometric sum is over primitive nonidentity <conjugacy classes> in $\Gamma$, equivalently \b[oriented] <primitive closed geodesics>; a class and its inverse are both counted. The inner sum records their iterates. The first term is the identity contribution, proportional to <hyperbolic area>. This formula follows by taking the Gaussian test function $h(r)=e^{-t(r^2+1/4)}$ in the usual spectral-parameter formula, whose <Fourier transform> is the displayed Gaussian factor.<Huber's theorem for hyperbolic orbisurfaces by Emily B. Dryden and Alexander Strohmaier>[Section 2, equation (1), specialized to a torsion-free group and a Gaussian test function]{ref}
The left side is the <heat trace>. Thus the same quantity can be computed from vibration frequencies or from closed-geodesic lengths and <hyperbolic area>. The <Huber spectral equivalence theorem> extracts the mutual determination of the two spectra. Quotients with cusps or cone points require additional terms, so the displayed version uses the closed smooth orientable setting.
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