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arXiv · 2205.05822

On a Tur\'an conjecture and random multiplicative functions

Rodrigo Angelo·Max Wenqiang Xu
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Abstract

We show that if $f$ is the random completely multiplicative function, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is positive for every $x$ is at least $1-10^{-45}$, while also strictly smaller than $1$. For large $x$, we prove an asymptotic upper bound of $O(\exp(-\exp( \frac{\log x}{C\log \log x })))$ on the exceptional probability that a particular truncation is negative, where $C$ is some positive constant.

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Rodrigo Angelo, Max Wenqiang Xu. 2022-05-12. On a Tur\'an conjecture and random multiplicative functions. https://arxiv.org/abs/2205.05822

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