Finding the distribution of random multiplicative functions in short intervals

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• Adam Harper (University of Warwick)
• Wednesday 11 February 2026, 13:30-14:30
• MR4, CMS.

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Let f be a random multiplicative function, and consider the sum of f(n) over a short interval x ≤ n ≤ x+y (where y = o(x) as x tends to infinity). Thanks to work of Chatterjee-Soundararajan and Soundararajan-Xu, it is known that these sums have a Gaussian limiting distribution when rescaled by their standard deviation, provided x/y is at least a certain power of log x. On the other hand, work of Harper and of Caich implies that these sums will converge to zero when rescaled by their standard deviation, if y is “close’’ to x. I will report on joint work (in preparation) of myself, Soundararajan and Xu on this problem. We find that on the full range y = o(x), the sums have a Gaussian limiting distribution when rescaled properly, but the correct scaling factor changes as y approaches x. In contrast, when y ~ x there is no rescaling under which the sums have a (non-degenerate) Gaussian limit.

This talk is part of the Discrete Analysis Seminar series.

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