We show that for a Steinhaus random multiplicative function \(f:{\mathbb {N}}\rightarrow {\mathbb {D}}\) and any polynomial \(P(x)\in {\mathbb {Z}}[x]\) of \(\deg P\ge 2\) which is not of the form \(w(x+c)^{d}\) for some \(w\in {\mathbb {Z}}\), \(c\in {\mathbb {Q}}\), we have

where \({{\mathcal {C}}}{{\mathcal {N}}}(0,1)\) 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

for any polynomial \(P(x)\in {\mathbb {Z}}[x]\) with \(\deg P\ge 2,\) 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 polynomial \(P(n)=n,\) 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 \(\varepsilon >0\).

我们证明，对于 Steinhaus 随机乘法函数\(f:{\mathbb {N}}\rightarrow {\mathbb {D}}\)和任何多项式\(P(x)\in {\mathbb {Z}}[ x]\)的\(\deg P\ge 2\)对于某些\(w\in {\mathbb {Z}}\)来说，其形式不是\(w(x+c)^{d}\) ) , \(c\in {\mathbb {Q}}\)，我们有

其中\({{\mathcal {C}}}{{\mathcal {N}}}(0,1)\)是均值为 0、方差为 1 的标准复高斯分布。这证实了 Najnudel 的猜想形式。我们进一步表明，几乎肯定存在任意大的值\(x\ge 1\)，使得

对于任何多项式\(P(x)\in {\mathbb {Z}}[x]\)与\(\deg P\ge 2,\) ，它不是线性因子的乘积（在\({\mathbb {Q}}\) )。这与迭代对数定律预测的界限相匹配。这两个结果都与众所周知的线性多项式\(P(n)=n,\)情况形成对比，其中已知部分和以非高斯方式表现，并且推测相应的急剧波动为\(O(\sqrt{x}(\log \log x)^{\frac{1}{4}+\varepsilon })\)对于任何\(\varepsilon >0\)。