A model problem for multiplicative chaos in number theory (Q2674713)

Let \((X(k))_{k\geq 1}\) be a sequence of independent standard complex Gaussians, and the sequence of random variables \((A(N))_{N\geq 0}\) defined by the formal power series identity \[ \exp\left(\sum_{k=1}^\infty\frac{X(k)}{\sqrt{k}}\,z^k\right)=\sum_{N=0}^\infty A(n)z^n. \] In this paper, the authors motivated by partial sums of random multiplicative functions, explain the fact that although \(\mathbb{E}[|A(N)|]\asymp\log^{-1/4}N\) as \(N\to\infty\), one has \(\mathbb{E}[|A(N)|^2]=1\) for all \(N\geq 1\).