Random Multiplicative Functions and Making Squares from Polynomial Values

Régis de la Bretèche; Victor Y. Wang; Max Wenqiang Xu

Random Multiplicative Functions and Making Squares from Polynomial Values

Number Theory 2026-07-07 v1 Probability

Authors: Régis de la Bretèche , Victor Y. Wang , Max Wenqiang Xu

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.

Cite

@article{arxiv.2607.06398,
  title  = {Random Multiplicative Functions and Making Squares from Polynomial Values},
  author = {Régis de la Bretèche and Victor Y. Wang and Max Wenqiang Xu},
  journal= {arXiv preprint arXiv:2607.06398},
  year   = {2026}
}

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24 pages