Random Multiplicative Functions and Making Squares from Polynomial Values

Abstract

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 \[P(n_1)P(n_2)P(n_3)P(n_4) = \square, \quad 1\le n_1, n_2, n_3, n_4 \le N.\] Results of Hooley, Evertse--Silverman, and Reuss play an important role in the proof. Our estimates are sharpest for $\hat{A}°P=2$, thanks to the rich theory of Pell--Fermat equations.

24 pages