发表机构
Université de Lorraine; CNRS（洛林大学; 法国国家科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对一类m周期复值函数族，改进了其部分和最大值的分布尾部估计并给出分布函数对数的渐近公式，还证明了该部分和最大值的结构定理，与Dirichlet特征和最大值分布的相关结果形成对比。
AI 中文摘要
在文献[ABL21]中，Autissier、Bonolis和Lamzouri在特定条件下得到了一类m周期复值函数族的部分和最大值的分布函数的一致估计，例如Kloosterman和或Birch和都满足这些条件。在本文中，我们在相同假设下，得到了这类族中部分和最大值的分布尾部的改进估计，首次在一个大的一致范围内给出了分布函数对数的渐近公式。此外，我们证明了这类复m周期函数族的部分和最大值的“结构定理”，该定理表明，在该族中普遍成立的是，大多数具有大范数的部分和接近其虚部，且最大值在m/2附近取得，这与Bober-Golmaker-Granville-Koukoulopoulos[BGGK18]、Lamzouri[Lam24]以及Lamzouri-Nath[LN24]关于各类Dirichlet特征族中特征和最大值分布的结果形成鲜明对比。
英文摘要
In \cite{ABL21}, Autissier, Bonolis and Lamzouri obtained uniform estimates for the distribution function of the maximum of partial sums of a class of families of $m$-periodic complex-valued functions under certain conditions. For example, these conditions are verified by Kloosterman or Birch sums. In this article, under the same assumptions, we obtain an improved estimate for the tail of the distribution of the maximum of partial sums in these families, which gives for the first time an asymptotic formula for the logarithm of the distribution function, in a large uniform range. Furthermore, we prove a "structure theorem" for the maximum of partial sums of a family of complex $m$-periodic functions in our class, which shows that universally over this class, most of the partial sums with large norm are close to their imaginary part, and the maximum is attained around $m/2$. This is in sharp constrast with the results of Bober-Golmaker-Granville-Koukoulopoulos \cite{BGGK18}, Lamzouri \cite{Lam24} and Lamzouri-Nath \cite{LN24} for the distribution of the maximum of character sums in various families of Dirichlet characters.
Comments38 pages
