An almost sure upper bound for random multiplicative functions on integers with a large prime factor

Daniele Mastrostefano

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arXiv · 2105.09565

An almost sure upper bound for random multiplicative functions on integers with a large prime factor

Daniele Mastrostefano

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Abstract

Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\ P(n)>\sqrt{x}}}f(n)\bigg|\leq\sqrt{x}(\log\log x)^{1/4+\varepsilon},$$ where $P(n)$ stands for the largest prime factor of $n$. This gives an indication of the almost sure size of the largest fluctuations of $f$.

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Daniele Mastrostefano. 2021-05-20. An almost sure upper bound for random multiplicative functions on integers with a large prime factor. https://arxiv.org/abs/2105.09565

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