Local square mean in the hyperbolic circle problem and sums of Salié sums
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914468982685696 |
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| author | Biró, András |
| author_facet | Biró, András |
| contents | Let $Γ\subseteq PSL(2, \mathbb R)$ 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 $e^{\frac 23R}$ is known, and this has not been improved for any group. Recently, taking $ z=w$ and considering $Γ= PSL(2, \mathbb Z)$, we have shown the estimate $ e^{\left(\frac 9{14}+ε\right)R}$ for the local $L^2$-norm of the error term, which is better than the pointwise bound. Here we improve the exponent $\frac 9{14}$, conditionally on a twisted Linnik-Selberg-type conjecture for sums of Salié sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11205 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local square mean in the hyperbolic circle problem and sums of Salié sums Biró, András Number Theory Let $Γ\subseteq PSL(2, \mathbb R)$ 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 $e^{\frac 23R}$ is known, and this has not been improved for any group. Recently, taking $ z=w$ and considering $Γ= PSL(2, \mathbb Z)$, we have shown the estimate $ e^{\left(\frac 9{14}+ε\right)R}$ for the local $L^2$-norm of the error term, which is better than the pointwise bound. Here we improve the exponent $\frac 9{14}$, conditionally on a twisted Linnik-Selberg-type conjecture for sums of Salié sums. |
| title | Local square mean in the hyperbolic circle problem and sums of Salié sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.11205 |