论文标题
几乎确定随机乘法函数的大波动
Almost sure large fluctuations of random multiplicative functions
论文作者
论文摘要
我们证明，如果$ f（n）$是steinhaus或rademacher随机乘法函数，则几乎肯定存在$ x $的任意大值，$ | \ sum_ {n \ leq x} f（n）| \ geq \ sqrt {x}（\ log \ log x）^{1/4+o（1）} $。这是第一个生长速度比$ \ sqrt {x} $更快的界限，回答了哈拉斯的问题并证明了Erdős的猜想。指数$ 1/4 $在此问题上很尖锐，这是合理的。 这些证明是通过在$ x $的序列上建立$ \ sum_ {n \ leq x} f（n）$的多元高斯近似值来起作用的，条件是所有最大的primes $ p $以外的所有行为。最困难的方面表明，总和的条件协方差通常很小，因此相应的高斯通常大致独立。这些协方差与由添加剂特征扭曲的Euler产品（或乘法混乱）类型相关，我们使用各种工具进行了研究，包括Dirichlet多项式的平均值估计值，随机Euler产品的高混合力矩估计以及随机步行的屏障参数。
We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum_{n \leq x} f(n)| \geq \sqrt{x} (\log\log x)^{1/4+o(1)}$. This is the first such bound that grows faster than $\sqrt{x}$, answering a question of Halász and proving a conjecture of Erdős. It is plausible that the exponent $1/4$ is sharp in this problem. The proofs work by establishing a multivariate Gaussian approximation for the sums $\sum_{n \leq x} f(n)$ at a sequence of $x$, conditional on the behaviour of $f(p)$ for all except the largest primes $p$. The most difficult aspect is showing that the conditional covariances of the sums are usually small, so the corresponding Gaussians are usually roughly independent. These covariances are related to an Euler product (or multiplicative chaos) type integral twisted by additive characters, which we study using various tools including mean value estimates for Dirichlet polynomials, high mixed moment estimates for random Euler products, and barrier arguments with random walks.
