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

2025/06/22 by Diao, Yijie
#11D57 #11G35) #11N32 (11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.18065

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 Châtelet 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-Thélène.

Related

• Bateman-Horn, polynomial Chowla and the Hasse principle with probability 1
• The distribution of prime values of random polynomials
• Averages of arithmetic functions over polynomials in many variables
• Local solubility in generalised Châtelet varieties
• Generalized Galois numbers, inversions, lattice paths, Ferrers diagrams\n and limit theorems