带周期权重的随机乘性函数
Random Multiplicative Functions with Periodic Weights
AI总结：
该研究探讨带1周期权重的Steinhaus随机乘性函数和的分布，得出归一化和收敛于标准复高斯分布的充要条件，并证明限制素因子大于趋于无穷的z时和始终满足中心极限定理。
AI中文摘要：
给定一个Steinhaus随机乘性函数$f$、一个有界变差的1周期函数$g$以及一个无理数$\alpha$，我们研究$\sum_{n=1}^N f(n) g(\alpha n)$的分布。我们确定了这些经标准差归一化的和收敛于标准复高斯分布的充要条件。另一方面，我们证明：若将求和项限制为所有素因子都大于$z$的数，且$z$以任意慢的速度趋于无穷，则这类和始终满足中心极限定理。
英文摘要：
Given a Steinhaus random multiplicative function $f$, a $1$-periodic function of bounded variation $g$, and an irrational number $α$, we study the distribution of $\sum_{n=1}^N f(n) g(αn)$. We determine a necessary and sufficient condition for these sums, normalized by their standard deviation, to converge to the standard complex Gaussian distribution. On the other hand, we show that if we restrict the summands to have all their prime factors $>z$, with $z$ tending to infinity arbitrarily slowly, then a central limit theorem always holds for such sums.
