The distribution of large values of mixed character sums
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908882115231744 |
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| author | Iggidr, Amine |
| author_facet | Iggidr, Amine |
| contents | In this paper, we investigate the distribution of values of the complete exponential sum $S_{p,χ}(θ)=\sum_{n=1}^p χ(n)e(nθ)$, where $p$ is a large prime, $χ$ is a Dirichlet character (mod $p$) of order $d\geq 2$, and $θ$ varies over certain subsets of $[0,1]$. When $d=2$, these sums correspond to the values of the Fekete polynomial associated with $p$ on the unit circle. Our first result gives precise estimates for the tail of the distribution of $|S_{p,χ}(θ)|$ in a large uniform range, when $θ$ varies over the set $\{(k+1/2)/p\}_{1\leq k\leq p}$. This improves upon a result of Conrey, Granville, Poonen, and Soundararajan. We also consider the distribution of the maximum of $|S_{p,χ}(θ)|$ for $θ\in I_k=[k/p,(k+1)/p]$, and obtain upper and lower bounds for the distribution of large values of this maximum, valid in a uniform range that is nearly optimal: we make this precise in the paper. Our results provide strong support for a conjecture of Montgomery on the maximum of Fekete polynomials on the unit circle. In particular, we show that the distribution function exhibits double-exponential decay, with a surprising difference in behavior between the cases of even and odd order $d$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_12159 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The distribution of large values of mixed character sums Iggidr, Amine Number Theory In this paper, we investigate the distribution of values of the complete exponential sum $S_{p,χ}(θ)=\sum_{n=1}^p χ(n)e(nθ)$, where $p$ is a large prime, $χ$ is a Dirichlet character (mod $p$) of order $d\geq 2$, and $θ$ varies over certain subsets of $[0,1]$. When $d=2$, these sums correspond to the values of the Fekete polynomial associated with $p$ on the unit circle. Our first result gives precise estimates for the tail of the distribution of $|S_{p,χ}(θ)|$ in a large uniform range, when $θ$ varies over the set $\{(k+1/2)/p\}_{1\leq k\leq p}$. This improves upon a result of Conrey, Granville, Poonen, and Soundararajan. We also consider the distribution of the maximum of $|S_{p,χ}(θ)|$ for $θ\in I_k=[k/p,(k+1)/p]$, and obtain upper and lower bounds for the distribution of large values of this maximum, valid in a uniform range that is nearly optimal: we make this precise in the paper. Our results provide strong support for a conjecture of Montgomery on the maximum of Fekete polynomials on the unit circle. In particular, we show that the distribution function exhibits double-exponential decay, with a surprising difference in behavior between the cases of even and odd order $d$. |
| title | The distribution of large values of mixed character sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.12159 |