论文标题
在随机的chowla猜想上
On the random Chowla conjecture
论文作者
论文摘要
我们表明，对于Steinhaus随机乘法函数$ f：\ mathbb {n} \ to \ mathbb {d} $和任何多项式$ p（x）\ in \ in \ at \ mathbb {z} [x] $ of $ \ text {ge} \ Mathbb {z} $，$ c \ in \ Mathbb {q} $，我们有\ [\ frac {1} {\ sqrt {x}} \ sum_ {n \ le x} f（n \ le x} f（p（n）） $ \ MATHCAL {CN}（0,1）$是标准的复杂高斯分布，平均$ 0 $和Variance $1。$这证实了Najnudel的猜想。我们进一步表明，几乎肯定存在$ x \ ge 1，$的任意大值，以至于$$ | \ sum_ {n \ le x} f（p（n））| \ gg _ {\ text {deg} \ p} \ sqrt {x}（\ log \ log \ log x）^{1/2}，$ $ $ \ mathbb {q} $）。这与迭代对数定律的预测相匹配。这两种结果都与众所周知的线性阶段$ p（n）= n，$形成鲜明对比，其中已知部分总和以非高斯的方式行为，相应的急剧波动被认为是$ o（\ sqrt {x} {x}} $ \ varepsilon> 0 $。
We show that for a Steinhaus random multiplicative function $f:\mathbb{N}\to\mathbb{D}$ and any polynomial $P(x)\in\mathbb{Z}[x]$ of $\text{deg}\ P\ge 2$ which is not of the form $w(x+c)^{d}$ for some $w\in \mathbb{Z}$, $c\in \mathbb{Q}$, we have \[\frac{1}{\sqrt{x}}\sum_{n\le x} f(P(n)) \xrightarrow{d} \mathcal{CN}(0,1),\] where $\mathcal{CN}(0,1)$ is the standard complex Gaussian distribution with mean $0$ and variance $1.$ This confirms a conjecture of Najnudel in a strong form. We further show that there almost surely exist arbitrary large values of $x\ge 1,$ such that $$|\sum_{n\le x} f(P(n))| \gg_{\text{deg}\ P} \sqrt{x} (\log \log x)^{1/2},$$ for any polynomial $P(x)\in\mathbb{Z}[x]$ with $\text{deg}\ P\ge 2,$ which is not a product of linear factors (over $\mathbb{Q}$). This matches the bound predicted by the law of the iterated logarithm. Both of these results are in contrast with the well-known case of linear phase $P(n)=n,$ where the partial sums are known to behave in a non-Gaussian fashion and the corresponding sharp fluctuations are speculated to be $O(\sqrt{x}(\log \log x)^{\frac{1}{4}+\varepsilon})$ for any $\varepsilon>0$.
