Short Sums of the Liouville Function over Function Fields
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866929665340342272 |
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| author | Fleet, Simon |
| author_facet | Fleet, Simon |
| contents | Let $λ$ denote the Liouville function for function fields. We prove that for a fixed $q$, given $h \ll \sqrt{N}$ and $h(N) \to \infty$ arbitrarily slowly as $N \to \infty$, then \begin{equation*} \frac{1}{q^N}\sum_{G_0 \in \mathcal{M}_N}|\sum_{G \in \mathcal{I}_{h}(G_0)}λ(G)|^2 \ll_q \frac{N^5}{h^2}q^{h}. \end{equation*} The proof follows a similar method of an analogous case in the integer setting developed by Chinis, adapting methods originally developed by Matomäki and Radziwiłł. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Short Sums of the Liouville Function over Function Fields Fleet, Simon Number Theory Let $λ$ denote the Liouville function for function fields. We prove that for a fixed $q$, given $h \ll \sqrt{N}$ and $h(N) \to \infty$ arbitrarily slowly as $N \to \infty$, then \begin{equation*} \frac{1}{q^N}\sum_{G_0 \in \mathcal{M}_N}|\sum_{G \in \mathcal{I}_{h}(G_0)}λ(G)|^2 \ll_q \frac{N^5}{h^2}q^{h}. \end{equation*} The proof follows a similar method of an analogous case in the integer setting developed by Chinis, adapting methods originally developed by Matomäki and Radziwiłł. |
| title | Short Sums of the Liouville Function over Function Fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.04461 |