论文标题
在短移动间隔中，字符总和的注释
A note on character sums over short moving intervals
论文作者
论文摘要
我们研究了总和$（1/\ sqrt {h}）\ sum_ {x <n \ leq x+h}χ（n）$，其中$χ$是固定的非主要dirichlet字符modulo a Prime $ q $，$ 0 \ leq x \ leq x \ leq x \ leq x $是均匀的。 Davenport和Erdős以及最近的Lamzouri证明了这些总和提供了$ H \ rightArrow \ rightarrow \ infty $和$（\ log h）/\ log q \ rightArrow 0 $ as $ q \ rightArrow \ rightArrow \ rightArrow \ infty \ infty $，以及Lamzouri推测的这些应该将受试者限制在$ eper oger $ hh $ h = o的$ h = o（q）。我们证明，即使对于任何固定$ a> 0 $的$ h = q/\ log^{a} q $，这对于某些$χ$来说是错误的。另一方面，我们证明了范围内的“几乎所有”字符$ q^{1-o（1）} \ leq h = o（q）$是正确的。 使用Pólya的傅立叶扩展，这些结果可能会重新构成，以说明具有数量理论系数的某些傅立叶序列分布的陈述。证明中使用的工具包括在短初始段中存在具有大量总和的字符，以及具有随机乘数系数的三角多项式的矩估计。
We investigate the sums $(1/\sqrt{H}) \sum_{X < n \leq X+H} χ(n)$, where $χ$ is a fixed non-principal Dirichlet character modulo a prime $q$, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided $H \rightarrow \infty$ and $(\log H)/\log q \rightarrow 0$ as $q \rightarrow \infty$, and Lamzouri conjectured these should hold subject to the much weaker upper bound $H=o(q/\log q)$. We prove this is false for some $χ$, even when $H = q/\log^{A}q$ for any fixed $A > 0$. On the other hand, we show it is true for "almost all" characters on the range $q^{1-o(1)} \leq H = o(q)$. Using Pólya's Fourier expansion, these results may be reformulated as statements about the distribution of certain Fourier series with number theoretic coefficients. Tools used in the proofs include the existence of characters with large partial sums on short initial segments, and moment estimates for trigonometric polynomials with random multiplicative coefficients.
