Spectral Exponential Sums on Hyperbolic Surfaces
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866913625648660480 |
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| author | Kaneko, Ikuya |
| author_facet | Kaneko, Ikuya |
| contents | We study an exponential sum over Laplacian eigenvalues $λ_{j} = 1/4+t_{j}^{2}$ with $t_{j} \leqslant T$ for Maass cusp forms on $Γ\backslash \mathbb{H}$, where $Γ$ is a cofinite Fuchsian group acting on the upper half-plane $\mathbb{H}$. The aim is to establish an asymptotic formula which expresses spectral exponential sums in terms of an oscillatory component, von Mangoldt-like functions and Selberg zeta functions. The behaviour is determined by whether $Γ$ is essentially cuspidal or not. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1905_00681 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Spectral Exponential Sums on Hyperbolic Surfaces Kaneko, Ikuya Number Theory Spectral Theory 11M36 (primary), 11F72 (secondary) We study an exponential sum over Laplacian eigenvalues $λ_{j} = 1/4+t_{j}^{2}$ with $t_{j} \leqslant T$ for Maass cusp forms on $Γ\backslash \mathbb{H}$, where $Γ$ is a cofinite Fuchsian group acting on the upper half-plane $\mathbb{H}$. The aim is to establish an asymptotic formula which expresses spectral exponential sums in terms of an oscillatory component, von Mangoldt-like functions and Selberg zeta functions. The behaviour is determined by whether $Γ$ is essentially cuspidal or not. |
| title | Spectral Exponential Sums on Hyperbolic Surfaces |
| topic | Number Theory Spectral Theory 11M36 (primary), 11F72 (secondary) |
| url | https://arxiv.org/abs/1905.00681 |