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 $°P = 2$, thanks to the rich theory of Pell--Fermat equations.

Disclosure

“B(N, N 1/2 ) ≪P,δ N 1−δ (9) for all N ⩾ 1, then we say that Hδ holds. The following result builds on work of Booker–Browning [9, §4]. We identified the proof strategy, and GPT-5.5 Pro assisted in drafting an initial version of the proof, with a worse exponent δ0 . We then checked and substantially revised the argument, especially in the following aspects: improving the treatment of the ranges I2 and I3 in [9, §4], a”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-24T00:34:12.480Z

Structural counts