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Mathematics > Number Theory

arXiv:2108.11367 (math)
[Submitted on 25 Aug 2021]

Title:A conjectural asymptotic formula for multiplicative chaos in number theory

Authors:Daksh Aggarwal, Unique Subedi, William Verreault, Asif Zaman, Chenghui Zheng
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Abstract:We investigate a special sequence of random variables $A(N)$ defined by an exponential power series with independent standard complex Gaussians $(X(k))_{k \geq 1}$. Introduced by Hughes, Keating, and O'Connell in the study of random matrix theory, this sequence relates to Gaussian multiplicative chaos (in particular "holomorphic multiplicative chaos'' per Najnudel, Paquette, and Simm) and random multiplicative functions. Soundararajan and Zaman recently determined the order of $\mathbb{E}[|A(N)|]$. By constructing an algorithm to calculate $A(N)$ in $O(N^2 \log N)$ steps, we produce computational evidence that their result can likely be strengthened to an asymptotic result with a numerical estimate for the asymptotic constant. We also obtain similar conclusions when $A(N)$ is defined using standard real Gaussians or uniform $\pm 1$ random variables. However, our evidence suggests that the asymptotic constants do not possess a natural product structure.
Comments: 18 pages
Subjects: Number Theory (math.NT); Probability (math.PR)
MSC classes: 60-08 (Primary), 60-11, 60F99, 11K65 (Secondary)
Cite as: arXiv:2108.11367 [math.NT]
  (or arXiv:2108.11367v1 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.2108.11367
arXiv-issued DOI via DataCite
Journal reference: Res. Number Theory 8, 35 (2022)
Related DOI: https://doi.org/10.1007/s40993-022-00332-x
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From: Daksh Aggarwal [view email]
[v1] Wed, 25 Aug 2021 17:55:13 UTC (1,554 KB)
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