Counting square-free values of random polynomials

TL;DR

The paper investigates the typical behavior of square-free values represented by integer polynomials by analyzing the average error term $E_P(x)=\sharp\{n\le x: P(n) \text{ square-free}\}-\mathfrak S_P x$ over random polynomials of fixed degree. It develops a random-model framework with a truncated sieve weight and a second-moment analysis that isolates an off-diagonal constant $\gamma_0$ and a correlation function $c(m,n)$, enabling a Cesàro-summed Perron integral to control divergent sums. By combining precise zeta-function bounds, contour-shifting techniques, and optimal truncation $z=H^{\lambda}$ with $\lambda=(d+1)/(4d+3)$, the authors extract a main-term asymptotic and prove that the average error is of size $O(x^{1/4})$ relative to the main term. This provides a rigorous probabilistic description of the typical fluctuations in the square-free values taken by random polynomials and establishes a framework for studying higher moments and broader random models in the future.

Abstract

We prove that the average error term when counting square-free values of polynomials is the quartic root of the main term.

Theorems & Definitions (32)

• Theorem 1.1
• Remark 1.2: Higher moments
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof