Abstract
A Steinhaus random multiplicative function f is a completely multiplicative function obtained by setting its values on primes f(p) to be independent random variables distributed uniformly on the unit circle. Recent work of Harper shows that \(\sum\nolimits_{n \le N} {f(n)} \) exhibits “more than square-root cancellation,” and in particular \({1 \over {\sqrt N }}\sum\nolimits_{n \le N} {f(n)} \) does not have a (complex) Gaussian distribution. This paper studies \(\sum\nolimits_{n \in {\cal A}} {f(n)} \), where \({\cal A}\) is a subset of the integers in [1, N], and produces several new examples of sets \({\cal A}\) where a central limit theorem can be established. We also consider more general sums such as \(\sum\nolimits_{n \le N} {f(n){e^{2\pi in\theta }}} \), where we show that a central limit theorem holds for any irrational θ that does not have extremely good Diophantine approximations.
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Acknowledgments
We thank Adam Harper for helpful discussions and comments on an earlier version of the paper. We are also grateful to Louis Gaudet for raising a question during the second author’s graduate student seminar at AIM, which led us to Corollary 1.2. Thanks are also due to the referee for a careful reading. K. S. is partially supported through a grant from the National Science Foundation, and a Simons Investigator Grant from the Simons Foundation. M. W. X. is partially supported by the Cuthbert C. Hurd Graduate Fellowship in the Mathematical Sciences, Stanford.
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To Peter Sarnak on the occasion of his seventieth birthday
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Soundararajan, K., Xu, M.W. Central limit theorems for random multiplicative functions. JAMA 151, 343–374 (2023). https://doi.org/10.1007/s11854-023-0331-y
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DOI: https://doi.org/10.1007/s11854-023-0331-y