Random Multiplicative Functions and Making Squares from Polynomial Values

For a large family of polynomials $P(X)\in \mathbb{Z}[X]$, we prove central limit theorems for ${\sum }_{n\le N}f(P(n))$ for both Rademacher and extended Rademacher multiplicative functions $f$. To achieve this, we establish a paucity phenomenon in counting solutions to $$P(n_1)P(n_2)P(n_3)P(n_4)=\mathrm{\square },1\le n_1,n_2,n_3,n_4\le N\mathrm{.}$$ Results of Hooley, Evertse–Silverman, and Reuss play an important role in the proof. Our estimates are sharpest for $\mathrm{°}P=2$, thanks to the rich theory of Pell–Fermat equations.