The distribution of prime values of random polynomials

TL;DR

This paper proves that almost all polynomials follow the Bateman--Horn and Hardy--Littlewood conjectures on prime values, and shows that the distribution of prime gaps and sign patterns of the Liouville function for these polynomials align with classical probabilistic models.

Contribution

It establishes that 100% of polynomials satisfy the Bateman--Horn and Hardy--Littlewood conjectures on average, and analyzes the distribution of prime gaps and sign patterns for these polynomials.

Findings

100% of polynomials satisfy the Bateman--Horn Conjecture in an $L^k$ sense.

100% of polynomials satisfy an analogue of the Hardy--Littlewood Prime Tuples Conjecture in an $L^2$ sense.

The distribution of prime gaps around the average spacing is Poisson for almost all 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 M\"obius functions (in turn relying on Leng, Sah, and Sawhney's quantitative inverse theorem for the Gowers norms).