The distribution of prime values of random polynomials

TL;DR

The paper studies the distribution of prime values taken by irreducible polynomials when the coefficients are averaged over in a high-dimensional, exponential-in-size regime. By turning the polynomial into a linear system in its coefficients for fixed inputs, the authors deploy Leng’s higher-order Fourier-analytic methods to obtain almost-all Bateman–Horn results, and to describe precise distributional behavior for prime counts and Liouville sign patterns. They prove that, on average, Bateman–Horn predictions hold to all moments, Hardy–Littlewood-type prime tuples along polynomials match in an averaged sense, and the tail distributions of prime gaps follow Poisson and Gaussian laws in appropriate regimes. A central theme is that substantial averaging unlocks quasirandomness phenomena for primes and Liouville values, with Leng’s uniformity results providing the crucial analytic backbone. The results give strong, nontrivial evidence toward the original (unaveraged) conjectures and illuminate the probabilistic structure underlying prime values of polynomials.

Abstract

The Bateman--Horn Conjecture predicts how often an irreducible polynomial $f(x) \in \mathbb{Z}[x]$ assumes prime values. We demonstrate that with sufficient averaging in the coefficients of $f$ (viz. exponential in the size of the inputs), one can not only prove Bateman--Horn results on average but also pin down precise information about the distribution of prime values. We show that 100\% of polynomials (in an $L^k$ sense for all $k \in \mathbb{N}$) satisfy the Bateman--Horn Conjecture, and that that 100\% of polynomials (in an $L^2$ sense) satisfy an appropriate polynomial analogue of the Hardy--Littlewood Prime Tuples Conjecture. We use the latter to prove that 100\% of polynomials satisfy the appropriate analogue of the Poisson Tail Conjecture, in the sense that the distribution of the gaps between consecutive prime values around the average spacing is Poisson. We also study the frequencies of sign patterns of the Liouville function evaluated at the consecutive outputs of $f$; viewing $f$ as a random variable, we establish the limiting distribution for every sign pattern. The Chowla problem along random polynomials is a special case. A key input behind all of our arguments is Leng's recent quantitative work on the higher-order Fourier uniformity of the von Mangoldt and Möbius functions (in turn relying on Leng, Sah, and Sawhney's quantitative inverse theorem for the Gowers norms).

Theorems & Definitions (37)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Theorem 1.10