论文标题
$ \ sum_ {n \ le x} \ frac {f（n）} {n} $ be有多负面？
How negative can $\sum_{n\le x}\frac{f(n)}{n}$ be?
论文作者
论文摘要
Turán观察到，对数部分总和$ \ sum_ {n \ le x} \ frac {f（n）} {n} {n} $完全乘法函数（在liouville函数的特定情况下$ f（n）=λ（n）$）往往是正的。我们开发了一种通用方法来证明旨在解释这种现象的两个结果。首先，我们表明，对于每$ \ varepsilon> 0 $，存在一些$ x_0 \ ge 1，$，使得对于任何完全乘法函数$ f $满足$ -1 \ le f（n）\ le 1 $，我们有$ \ sum_ \ sum_ \ sum_ \ sum_ {n \ le x}} \ frac {f frac \ frac {f（n） - \ frac {1} {（\ log \ log {x}）^{1- \ varepsilon}}，\ quad x \ ge x_0。$ $$这改善了由于Granville和soundararajan的限制。其次，我们表明，如果$ f $是典型的（随机）完全乘法函数$ f：\ mathbb {n} \ to \ { - { - 1,1 \} $，$ \ sum_ {n \ le x} \ le x} \ frac \ frac {f frac {f（n）} $是负面的概率。 $（\ exp（ - \ exp（\ frac {\ log x \ cdot \ log \ log \ log \ log x}} {c \ log \ log x}）））。
Turán observed that logarithmic partial sums $\sum_{n\le x}\frac{f(n)}{n}$ of completely multiplicative functions (in the particular case of the Liouville function $f(n)=λ(n)$) tend to be positive. We develop a general approach to prove two results aiming to explain this phenomena. Firstly, we show that for every $\varepsilon>0$ there exists some $x_0\ge 1,$ such that for any completely multiplicative function $f$ satisfying $-1\le f(n)\le 1$, we have $$\sum_{n\le x}\frac{f(n)}{n}\ge -\frac{1}{(\log\log{x})^{1-\varepsilon}}, \quad x\ge x_0.$$ This improves a previous bound due to Granville and Soundararajan. Secondly, we show that if $f$ is a typical (random) completely multiplicative function $f:\mathbb{N}\to \{-1,1\}$, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is negative for a given large $x,$ is $O(\exp(-\exp(\frac{\log x\cdot \log\log\log x}{C\log \log x}))).$ This improves on recent work of Angelo and Xu.
