2,900 research outputs found

    The prime geodesic theorem for PSL2(Z[i])\mathrm{PSL}_{2}(\mathbb{Z}[i]) and spectral exponential sums

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    We shall ponder the Prime Geodesic Theorem for the Picard manifold M=PSL2(Z[i])\h3\mathcal{M} = \mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathfrak{h}^{3}, which asks about the asymptotic behaviour of a counting function for the closed geodesics on M\mathcal{M}. Let EΓ(X)E_{\Gamma}(X) be the error term arising from counting prime geodesics, we then prove the bound EΓ(X)X3/2+ϵE_{\Gamma}(X) \ll X^{3/2+\epsilon} on average, as well as various versions of pointwise bounds. The second moment bound is the pure counterpart of work of Balog et al. for Γ=PSL2(Z)\Gamma = \mathrm{PSL}_{2}(\mathbb{Z}), and the main innovation entails the delicate analysis of sums of Kloosterman sums with an explicit evaluation of oscillatory integrals. Our pointwise bounds concern Weyl-type subconvex bounds for quadratic Dirichlet LL-functions over Q(i)\mathbb{Q}(i). Interestingly, we are also able to establish an asymptotic law for the spectral exponential sum in the spectral aspect for a cofinite Kleinian group Γ\Gamma. Finally, we produce numerical experiments of its behaviour, visualising that EΓ(X)E_{\Gamma}(X) obeys a conjectural bound of the size O(X1+ϵ)O(X^{1+\epsilon}).Comment: Numerous improvements to the exposition; improved the quality of the main theorem (Theorem 1.1) and achieved additional theorems such as Theorems 1.4, 3.17, 4.1, and 5.

    Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

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    For Γ\Gamma a cofinite Kleinian group acting on H3\mathbb{H}^3, we study the Prime Geodesic Theorem on M=Γ\H3M=\Gamma \backslash \mathbb{H}^3, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on MM. Let EΓ(X)E_{\Gamma}(X) be the error in the counting of prime geodesics with length at most logX\log X. For the Picard manifold, Γ=PSL(2,Z[i])\Gamma=\mathrm{PSL}(2,\mathbb{Z}[i]), we improve the classical bound of Sarnak, EΓ(X)=O(X5/3+ϵ)E_{\Gamma}(X)=O(X^{5/3+\epsilon}), to EΓ(X)=O(X13/8+ϵ)E_{\Gamma}(X)=O(X^{13/8+\epsilon}). In the process we obtain a mean subconvexity estimate for the Rankin-Selberg LL-function attached to Maass-Hecke cusp forms. We also investigate the second moment of EΓ(X)E_{\Gamma}(X) for a general cofinite group Γ\Gamma, and show that it is bounded by O(X16/5+ϵ)O(X^{16/5+\epsilon}).Comment: Corrected proof of Theorem 3.3 (with a weaker bound), added two authors, 18 page
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