arXiv:2611.14579v1 Announce Type: new Abstract: We prove that for a Rademacher random multiplicative function $(f(n))_{n\in\mathbb{N}}$ and any non-linear polynomial $P\in\mathbb{Z}[x]$ admitting a positive constant $\kappa_P$ such that \[\lim_{N\rightarrow\infty} \frac{\#\{n\in\left[N\right]:P(n)\text{ is square-free}\}}{N} =\kappa_P,\] we have \[ \frac{1}{\sqrt{\kappa_PN}}\sum_{n\leqslant N}f(P(n)) \xrightarrow[]{d}\mathcal{N}(0,1), \] as $N\rightarrow\infty $, where convergence is in distribution to a standard Gaussian. Using a direct combinatorial argument, we establish paucity estimates for non-trivial square products of square-free polynomial values at every fixed moment order. These estimates yield the Gaussian limit by the method of moments. We also study partial sums of the form $\sum_{n<N} F(f(n),f(N-n))$ for a suitable class of bivariate functions $F$ when $f$ is a Steinhaus random multiplicative function, and prove analogous results in special cases in the Rademacher setting. These results can be viewed as random analogues of the Goldbach-type problems considered by Mangerel.
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