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arXiv · 2303.14682

Sign changes of the partial sums of a random multiplicative function II

Marco Aymone
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Abstract

We study two models of random multiplicative functions: Rademacher random multiplicative functions supported on the squarefree integers $f$, and Rademacher random completely multiplicative functions $f^*$. We prove that the partial sums $\sum_{n\leq x}f^*(n)$ and $\sum_{n\leq x}\frac{f(n)}{\sqrt{n}}$ change sign infinitely often as $x\to\infty$, almost surely. The case $\sum_{n\leq x}\frac{f^*(n)}{\sqrt{n}}$ is left as an open question and we stress the possibility of only a finite number of sign changes, with positive probability.

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Marco Aymone. 2023-03-26. Sign changes of the partial sums of a random multiplicative function II. https://arxiv.org/abs/2303.14682

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