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

TL;DR

This paper studies the average behavior of arithmetic functions at polynomial values, establishing probabilistic versions of key conjectures and quantifying error terms with high precision.

Contribution

It introduces averaged versions of the Bateman-Horn and Chowla conjectures and analyzes the Hasse principle for norm form equations with explicit error bounds.

Findings

Proves averaged Bateman-Horn conjecture with probability 1.

Establishes averaged polynomial Chowla conjecture.

Quantifies the size of exceptional sets with logarithmic savings.

Abstract

With probability 1, we assess the average behaviour of various arithmetic functions at the values of degree d polynomials f 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.