Local square mean in the hyperbolic circle problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909036637585408 |
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| author | Biró, András |
| author_facet | Biró, András |
| contents | Let $Γ\subseteq PSL_2({\bf R})$ be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $Γ$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and $w$ are given points of the upper half plane and $R$ is a large number. An estimate with error term $e^{{2\over 3}R}$ is known, and this has not been improved for any group. Petridis and Risager proved that in the special case $Γ=PSL_2({\bf Z})$ taking $z=w$ and averaging over $z$ locally the error term can be improved to $e^{\left({7\over {12}}+ε\right)R}$. Here we show such an improvement for the local $L^2$-norm of the error term. Our estimate is $e^{\left({9\over {14}}+ε\right)R}$, which is better than the pointwise bound $e^{{2\over 3}R}$ but weaker than the bound of Petridis and Risager for the local average. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16113 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local square mean in the hyperbolic circle problem Biró, András Number Theory Let $Γ\subseteq PSL_2({\bf R})$ be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $Γ$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and $w$ are given points of the upper half plane and $R$ is a large number. An estimate with error term $e^{{2\over 3}R}$ is known, and this has not been improved for any group. Petridis and Risager proved that in the special case $Γ=PSL_2({\bf Z})$ taking $z=w$ and averaging over $z$ locally the error term can be improved to $e^{\left({7\over {12}}+ε\right)R}$. Here we show such an improvement for the local $L^2$-norm of the error term. Our estimate is $e^{\left({9\over {14}}+ε\right)R}$, which is better than the pointwise bound $e^{{2\over 3}R}$ but weaker than the bound of Petridis and Risager for the local average. |
| title | Local square mean in the hyperbolic circle problem |
| topic | Number Theory |
| url | https://arxiv.org/abs/2403.16113 |