A law of the iterated logarithm for random multiplicative functions on integers with a large prime factor

arXiv:2610.40707v1 Announce Type: new Abstract: Let $f$ be a Rademacher random multiplicative function supported on the squarefree integers. We investigate the almost sure fluctuations of the restricted sum $M^{>}(x)=\sum_{n\le x;~ P(n)>x^{1/2}}f(n)$, where $P(n)$ is the largest prime factor of $n$. With $Q(x)$ being the conditional variance of $M^{>}(x)$ given the values of $f$ at primes not exceeding $x^{1/2}$, we prove that, almost surely, the upper and lower limits of $M^{>}(x)(2Q(x)\log\log Q(x))^{-1/2}$ as $x\to\infty$ are $1$ and $-1$, respectively. Furthermore, it is shown that the almost sure cluster set of the normalized sum is $[-1,1]$.

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