The prime geodesic theorem for PSL2(ℤ[i]) and spectral exponential sums
نویسندگان
چکیده
This work addresses the Prime Geodesic Theorem for Picard manifold $\mathcal{M} = \mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathfrak{h}^{3}$, which asks asymptotic evaluation of a counting function closed geodesics on $\mathcal{M}$. Let $E_{\Gamma}(X)$ be error term in Theorem. We establish that $E_{\Gamma}(X) O_{\varepsilon}(X^{3/2+\varepsilon})$ average as well many pointwise bounds. The second moment bound parallels an analogous result $\Gamma \mathrm{PSL}_{2}(\mathbb{Z})$ due to Balog et al. and our innovation features delicate analysis sums Kloosterman with explicit manipulation oscillatory integrals. proof bounds requires Weyl-strength subconvexity quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Moreover, formula spectral exponential sum aspect cofinite Kleinian group $\Gamma$ is given. Our numerical experiments visualise particular obeys conjectural size $O_{\epsilon}(X^{1+\varepsilon})$.
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ژورنال
عنوان ژورنال: Algebra & Number Theory
سال: 2022
ISSN: ['1944-7833', '1937-0652']
DOI: https://doi.org/10.2140/ant.2022.16.1845