On the failure of weak convergence for Rademacher random multiplicative functions

Abstract: Let $\cP$ be the set of prime numbers, and $X_p,\, p\in\cP$ be a sequence of independent random variables such that $\bbP(X_p=\pm 1)=1/2$. Let $(\te_j){j=1}^{\infty}$ the corresponding random multiplicative functions of Radamacher type, namely, $\te_j=\prod{p|j}X_p$ if $j$ is square free and $\te_j=0$ otherwise. The motivation behind considering these variables comes from the Riemann Hypothesis since $(\te_\ell)$ can be viewed as the random counterpart of the M\"obius function $\mu(\ell)$ and the RH is equivalent to $\sum_{\ell\leq n}\mu(\ell)=o(n^{1/2+\varepsilon}), \forall \varepsilon>0$. Denote $S_n=\sum_{\ell=1}^n\te_\ell$. It is a natural guiding conjecture that $S_n/\sqrt n$ obeys the central limit theorem (CLT). However, S. Chatterjee conjectured (as expressed in \cite{[25]}) that the CLT should not hold. Chatterjee's conjecture was proved by Harper \cite{[17]}, and by now it is a direct consequence of a more recent breakthrough by Harper \cite{Har20} that $\frac{S_n}{\sqrt n}\to 0$ in distribution. Nevertheless, the question whether there exists a sequence $a_n=o(\sqrt n)$ such that $S_n/a_n$ converges to some limit remains a mystery. In this paper we, in particular, show that the answer is negative. Our proof is based on trigonometric identities, Levi's continuity theorem, the prime numbers theorem and the growth rates of the prime omega function $\om(n)$. Combining these tools we show that if a weak limit exists then the characteristic function of $S_n/a_n$ must converge to $0$ at any nonzero point. As a byproduct of our method we are also able to provide essentially optimal upper bounds on high order moments similar to \cite{Har19}, using different methods and for more general $X_p$'s.