Liouville function, von Mangoldt function and norm forms at random binary forms

TL;DR

This paper investigates the average behavior of arithmetic functions at values of binary forms, providing probabilistic versions of major conjectures and establishing the rational Hasse principle for certain Châtelet varieties.

Contribution

It introduces an average-case analysis of arithmetic functions at binary forms and proves an average version of Colliot-Thélène's conjecture for Châtelet varieties.

Findings

Averaged versions of Chowla and Bateman-Horn conjectures for random binary forms.

Rational Hasse principle holds for almost all Châtelet varieties under specified conditions.

Proof of an average version of Colliot-Thélène's conjecture.

Abstract

We analyze the average behavior of various arithmetic functions at the values of degree $d$ binary forms ordered by height, with probability $1$. This approach yields averaged versions of the Chowla conjecture and the Bateman-Horn conjecture for random binary forms. Furthermore, we show that the rational Hasse principle holds for almost all Ch\^atelet varieties defined by a fixed norm form of degree $e$ and by varying binary forms of fixed degree $d$, provided $e$ divides $d$. This proves an average version of a conjecture of Colliot-Th\'el\`ene.