arXiv 2202.087672022-02-21math.NTmath.PR

关于随机Chowla猜想

On the random Chowla conjecture

Oleksiy Klurman, Ilya D. Shkredov, Max Wenqiang Xu

备注：: Minor changes to the Introduction. Added remark about possibility of extending the proofs to sparse sets. Added references to the work of Cassaigne-Ferenczi-Mauduit-Rivat-Sarkozy and Borwein-Choi-Ganguly on the sign changes of $λ(P(n)).$ All other sections remain unchanged

摘要：

我们表明，对于一个Steinhaus随机乘法函数$f:\mathbb{N}\to\mathbb{D}$和任何多项式$P(x)\in\mathbb{Z}[x]$的$\text{deg}\ P\ge 2$这是不是的形式$w(x+c)^{d}$的一些$w\in \mathbb{Z}$，$c\in \mathbb{Q}$，我们有\[\frac{1}{\sqrt{x}}\sum_{n\le x} f(P(n)) \xrightarrow{d} \mathcal{CN}(0,1),\]其中$\mathcal{CN}(0,1)$是标准的复杂的高斯分布与平均$0$和方差$1.$这证实了猜想的Najnudel在一个强的形式。我们进一步表明，几乎肯定存在任意大值的$x\ge 1,$，使$$|\sum_{n\le x} f(P(n))| \gg_{\text{deg}\ P} \sqrt{x} (\log \log x)^{1/2},$$的任何多项式$P(x)\in\mathbb{Z}[x]$与$\text{deg}\ P\ge 2,$这不是一个产品的线性因素（在$\mathbb{Q}$）。这与重对数定律预测的界限相匹配。这两个结果是在对比与众所周知的情况下，线性相位$P(n)=n,$的部分和是已知的表现在一个非高斯的方式和相应的急剧波动被推测为$O(\sqrt{x}(\log \log x)^{\frac{1}{4}+\varepsilon})$的任何$\varepsilon>0$。

英文摘要：

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 $\varepsilon>0$.

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摘要：

英文摘要：

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