Central limit theorems for random multiplicative functions

Kannan Soundararajan, Max Wenqiang Xu
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Abstract

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 \(\sum\nolimits_{n \le N} {f(n)} \) exhibits “more than square-root cancellation,” and in particular \({1 \over {\sqrt N }}\sum\nolimits_{n \le N} {f(n)} \) does not have a (complex) Gaussian distribution. This paper studies \(\sum\nolimits_{n \in {\cal A}} {f(n)} \), where \({\cal A}\) is a subset of the integers in [1, N], and produces several new examples of sets \({\cal A}\) where a central limit theorem can be established. We also consider more general sums such as \(\sum\nolimits_{n \le N} {f(n){e^{2\pi in\theta }}} \), where we show that a central limit theorem holds for any irrational θ that does not have extremely good Diophantine approximations.

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随机乘法函数的中心极限定理
斯坦豪斯随机乘法函数 f 是一个完全乘法函数,它是通过将其在素数 f(p) 上的值设置为均匀分布在单位圆上的独立随机变量而得到的。哈珀(Harper)的最新研究表明,\(\sum\nolimits_{n \le N} {f(n)}\)表现出 "超过平方根的取消",尤其是\({1 over {\sqrt N }}\sum\nolimits_{n \le N} {f(n)}\)不具有(复)高斯分布。本文研究了 ( (sum\nolimits_{n \in {cal A}}{其中 \({\cal A}\) 是[1,N]中整数的子集,并产生了几个可以建立中心极限定理的集合 \({\cal A}\) 的新例子。我们还考虑了更一般的和,如(sum\nolimits_{n \le N} {f(n){e^{2\pi in\theta }}} )。\),在这里我们证明了中心极限定理对于任何无理数θ都是成立的,而这些无理数θ并没有极好的戴奥芬汀近似值。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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