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Squarefree values of random polynomials are Gaussian mixtures
Abstract
Fix d≥1 and choose a polynomial P uniformly at random from the set of integer polynomials of degree d whose coefficients have absolute value at most H. We study EP, the error in the count of squarefree values P(n) for n≤ x. In the range x=Ho(1), we prove that EP(x)/x1/4 converges to a Gaussian mixture that is not itself Gaussian, √(C(G))Z, where G is the weak limit, as H→∞, of the prime-indexed vector (Pbmod p2)p, C is a nonnegative function and Z is an independent standard Gaussian. This answers a question of Sofos. Finally, in the appendix, we establish that log 𝔼[C(G)m]asympd m2/(log m)2 as m→∞.
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BibTeX
@misc{simhi2026squarefree,
title = {{Squarefree values of random polynomials are Gaussian mixtures}},
author = {Omer Simhi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29911},
eprint = {2610.29911},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29911},
}
RIS
TY - UNPB
TI - Squarefree values of random polynomials are Gaussian mixtures
AU - Omer Simhi
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29911
UR - https://arxiv.org/abs/2610.29911
AB - Fix d≥1 and choose a polynomial P uniformly at random from the set of integer polynomials of degree d whose coefficients have absolute value at most H. We study E_P, the error in the count of squarefree values P(n) for n≤ x. In the range x=H^o(1), we prove that E_P(x)/x^1/4 converges to a Gaussian mixture that is not itself Gaussian, √(C(G))Z, where G is the weak limit, as H→∞, of the prime-indexed vector (Pbmod p^2)_p, C is a nonnegative function and Z is an independent standard Gaussian. This answers a question of Sofos. Finally, in the appendix, we establish that log 𝔼[C(G)^m]asymp_d m^2/(log m)^2 as m→∞.
ER -