Random Multiplicative Functions and Making Squares from Polynomial Values
Abstract
For a large family of polynomials P(X)∈ Z[X], we prove central limit theorems for Σn 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 \[P(n1)P(n2)P(n3)P(n4) = , 1 n1, n2, n3, n4 N.\] 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.
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