A conjectural asymptotic formula for multiplicative chaos in number theory

TL;DR

This paper explores the asymptotic behavior of a sequence of random variables linked to Gaussian multiplicative chaos, providing computational evidence for a conjectural formula and analyzing the constants involved.

Contribution

It introduces an algorithm for calculating the sequence efficiently and offers numerical evidence supporting a conjectural asymptotic formula for its expected magnitude.

Findings

Algorithm computes A(N) in O(N^2 log N) steps.

Numerical evidence supports a conjectural asymptotic formula.

Asymptotic constants likely lack a natural product structure.

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.