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Counting sign changes of partial sums of random multiplicative functions
Nick Geis, Ghaith Hiary · 2023-11-28 · via math.PR updates on arXiv.org

Let $f$ be a Rademacher random multiplicative function. Let $$M_f(u):=\sum_{n \leq u} f(n)$$ be the partial sum of $f$. Let $V_f(x)$ denote the number of sign changes of $M_f(u)$ up to $x$. We show that for any constant $c > 2$, $$V_f(x) = Ω((\log \log \log x)^{1/c} )$$ almost surely.

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