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arXiv · 2606.29040

Distribution of random multiplicative functions in short intervals, with proper normalization

Adam J. Harper·Kannan Soundararajan·Max Wenqiang Xu
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Abstract

We determine the limiting distribution of partial sums of a Steinhaus random multiplicative function $\sum_{x\le n \le x+y} f(n)$ over short intervals $[x, x+y]$, where $y \rightarrow \infty$ but $y=o(x)$. We show that with appropriate normalization, the limiting distribution is Gaussian for all such $y$. A key new feature of our result is that the normalization factor is different from the standard deviation $\sqrt{y}$ when $y$ is very close to $x$. In contrast, when $y \asymp x$ there is no normalization for which the limiting distribution is a non-degenerate Gaussian.

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Adam J. Harper, Kannan Soundararajan, Max Wenqiang Xu. 2026-06-27. Distribution of random multiplicative functions in short intervals, with proper normalization. https://arxiv.org/abs/2606.29040

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