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arXiv · 2607.06398

Random Multiplicative Functions and Making Squares from Polynomial Values

Régis de la Bretèche·Victor Y. Wang·Max Wenqiang Xu
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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 $\deg P = 2$, thanks to the rich theory of Pell--Fermat equations.

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Régis de la Bretèche, Victor Y. Wang, Max Wenqiang Xu. 2026-07-07. Random Multiplicative Functions and Making Squares from Polynomial Values. https://arxiv.org/abs/2607.06398

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