A lower bound on high moments of character sums
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913509798838272 |
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| author | Szabó, Barnabás |
| author_facet | Szabó, Barnabás |
| contents | For any real $k\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum
\begin{equation*}
\frac{1}{ϕ(q)} \sum_{\substack{χ\text{ mod }q\\ χ\neq χ_0}} \Big| \sum_{n\leq x} χ(n)\Big|^{2k}, \end{equation*} where $1\leq x\leq q$, and $χ$ is summed over the set of non-trivial Dirichlet characters mod $q$. Our bound is known to be optimal up to a constant factor under the Generalised Riemann Hypothesis. We also get a sharp lower bound on moments of theta functions using the same method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13436 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A lower bound on high moments of character sums Szabó, Barnabás Number Theory For any real $k\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum \begin{equation*} \frac{1}{ϕ(q)} \sum_{\substack{χ\text{ mod }q\\ χ\neq χ_0}} \Big| \sum_{n\leq x} χ(n)\Big|^{2k}, \end{equation*} where $1\leq x\leq q$, and $χ$ is summed over the set of non-trivial Dirichlet characters mod $q$. Our bound is known to be optimal up to a constant factor under the Generalised Riemann Hypothesis. We also get a sharp lower bound on moments of theta functions using the same method. |
| title | A lower bound on high moments of character sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.13436 |