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The distribution of partial sums of random multiplicative functions with a large prime factor
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abstract
For $f$ a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums \[ \frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{\substack{n \leq x \\ P(n) > \sqrt{x}}} f(n), \] where $P(n)$ denotes the largest prime factor of $n$. We find that the limiting distribution is given by the square root of an integral with respect to a critical Gaussian multiplicative chaos measure multiplied by an independent standard complex normal random variable.
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A sharp almost sure upper bound for partial sums of random multiplicative functions
For both Steinhaus and Rademacher random multiplicative functions, almost surely |sum_{n≤x} f(n)| ≪ sqrt(x)(log log x)^{1/4+ε}, matching Harper's lower bound.
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