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.
Fields of Study & Subjects
Analytic Number Theory Research
Advanced Mathematical Identities
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- Journal für die reine und angewandte Mathematik (Crelles Journal)
- Publisher
- Walter de Gruyter GmbH
- Publication date
- 24 August 2026
- Language
- EN