Bateman–Horn, polynomial Chowla and the Hasse principle with probability 1

Abstract

Abstract With probability 1, we assess the average behaviour of various arithmetic functions at the values of degree 𝑑 polynomials f ∈ Z ⁢ [ t ] f\in\mathbb{Z}[t] that are ordered by height. This allows us to establish averaged versions of the Bateman–Horn conjecture, the polynomial Chowla conjecture and to address a basic question about the integral Hasse principle for norm form equations. Moreover, we are able to quantify the error term in the asymptotics and the size of the exceptional set of 𝑓, both with arbitrary logarithmic power savings.