论文标题
在$ n $和$ \ lfloorαn\ rfloor $之间的乘法独立性上
On the multiplicative independence between $n$ and $\lfloor αn\rfloor$
论文作者
论文摘要
在本文中，我们调查了序列$ n $和$ \ lfloornα\ rfloor $之间的不同形式的乘法独立性，用于非理性$α$。我们的主要定理表明，对于一大批算术函数，$ a，b \ colon \ mathbb {n} \ to \ mathbb {c} $序列$（a（n））_ {n \ in \ mathbb {n}}}} $ and $（\ lfloor and $ rfloor -n \ rfloor -rfloor -rforor）in \ mathbb {n}} $渐近不相关。然后，将此新定理用于证明ERDőS-KAC定理的$ 2 $二维版本，并断言序列$（ω（n））_ {n \ in \ in \ Mathbb {n}}} $ and $（ω（ω）（ω（\ lfloorαn\ rfloer）正态分布的随机变量具有平均$ \ log \ log n $和标准偏差$ \ sqrt {\ log \ log n} $。 \ mathbb {n}} $倾向于$ 0 $。
In this article we investigate different forms of multiplicative independence between the sequences $n$ and $\lfloor n α\rfloor$ for irrational $α$. Our main theorem shows that for a large class of arithmetic functions $a, b \colon \mathbb{N} \to \mathbb{C}$ the sequences $(a(n))_{n \in \mathbb{N}}$ and $(b ( \lfloor αn \rfloor))_{n \in \mathbb{N}}$ are asymptotically uncorrelated. This new theorem is then applied to prove a $2$-dimensional version of the Erdős-Kac theorem, asserting that the sequences $(ω(n))_{n \in \mathbb{N}}$ and $(ω( \lfloor αn \rfloor)_{n\in \mathbb{N}}$ behave as independent normally distributed random variables with mean $\log\log n$ and standard deviation $\sqrt{ \log \log n}$. Our main result also implies a variation on Chowla's Conjecture asserting that the logarithmic average of $(λ(n) λ( \lfloor αn \rfloor))_{n \in \mathbb{N}}$ tends to $0$.
