论文标题
随机乘法函数的中心限制定理
Central limit theorems for random multiplicative functions
论文作者
论文摘要
Steinhaus随机乘法函数$ F $是通过将其值设置在Primes $ f（p）$上的独立随机变量均匀分布在单位圆上的独立随机变量，从而获得了完全乘法函数。哈珀的最新工作表明，$ \ sum_ {n \ le n} f（n）$展示了``超过平方根的取消''，尤其是$ \ frac 1 {\ sqrt {n}} \ sum_ sum_ sum_ {n \ le n} f（n \ le n} f（n）$ compers coper coper compears $ um_ conspect。 {\ Mathcal a}} f（n）$，其中$ {\ Mathcal a} $是$ [1，n] $中的整数的一个子集，并且可以建立$ \ sum_的中心限制，从而产生了几个集合$ {\ Mathcal a} $的新示例。 nθ} $，在其中我们表明，中央限制定理对于任何没有极好的二聚体近似值的非理性$θ$都保留。
A Steinhaus random multiplicative function $f$ is a completely multiplicative function obtained by setting its values on primes $f(p)$ to be independent random variables distributed uniformly on the unit circle. Recent work of Harper shows that $\sum_{n\le N} f(n)$ exhibits ``more than square-root cancellation," and in particular $\frac 1{\sqrt{N}} \sum_{n\le N} f(n)$ does not have a (complex) Gaussian distribution. This paper studies $\sum_{n\in {\mathcal A}} f(n)$, where ${\mathcal A}$ is a subset of the integers in $[1,N]$, and produces several new examples of sets ${\mathcal A}$ where a central limit theorem can be established. We also consider more general sums such as $\sum_{n\le N} f(n) e^{2πi nθ}$, where we show that a central limit theorem holds for any irrational $θ$ that does not have extremely good Diophantine approximations.
