On the distribution of very short character sums
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912743762690048 |
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| author | Nosal, Paweł |
| author_facet | Nosal, Paweł |
| contents | We establish a central limit theorem of $(1/\sqrt{h_p})\sum_{X< n \leq X+h_p}\big(\tfrac{n}{p}\big)$ for almost all the primes $p$, with $X$ uniformly random in $[g(p)]$, $g(p)$ an arbitrary divergent function growing slower than any power of $p$, provided $(\log h_p)/(\log g(p))\rightarrow 0, \, h_p \rightarrow \infty$ as $p \rightarrow \infty$. This improves the recent results of Basak, Nath and Zaharescu, who established this for $g(p) = (\log p)^A, A>1$. We also use the best currently available tools to expand the original central limit theorem of Davenport and Erdős for all the primes to a shorter interval of starting points.
In this paper we exploit a Selberg's sieve argument, recently used by Harper, an intersection result due to Evertse and Silverman and some consequences of the Weil bound on general character sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02915 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the distribution of very short character sums Nosal, Paweł Number Theory Probability We establish a central limit theorem of $(1/\sqrt{h_p})\sum_{X< n \leq X+h_p}\big(\tfrac{n}{p}\big)$ for almost all the primes $p$, with $X$ uniformly random in $[g(p)]$, $g(p)$ an arbitrary divergent function growing slower than any power of $p$, provided $(\log h_p)/(\log g(p))\rightarrow 0, \, h_p \rightarrow \infty$ as $p \rightarrow \infty$. This improves the recent results of Basak, Nath and Zaharescu, who established this for $g(p) = (\log p)^A, A>1$. We also use the best currently available tools to expand the original central limit theorem of Davenport and Erdős for all the primes to a shorter interval of starting points. In this paper we exploit a Selberg's sieve argument, recently used by Harper, an intersection result due to Evertse and Silverman and some consequences of the Weil bound on general character sums. |
| title | On the distribution of very short character sums |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2512.02915 |