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

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.