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

2026/07/21 by Yijie Diao

paper · doi:10.1017/s0017089526101074

crossref issued 2026/07/21 · crossref published 2026/07/21 · crossref published-online 2026/07/21 · crossref created 2026/07/21 · crossref deposited 2026/07/21 · crossref indexed 2026/07/30

Abstract

Abstract We analyze the average behavior of various arithmetic functions at the values of degree d d d binary forms ordered by height, with probability 1 1 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 e e and by varying binary forms of fixed degree d d d , provided e e e divides d d d . This proves an average version of a conjecture of Colliot-Thélène.

Citations

• Rational points and prime values of polynomials in moderately many\n variables
• The logarithmically averaged Chowla and Elliott conjectures for two-point correlations
• Primes represented by incomplete norm forms
• Failures of weak approximation in families
• Primes values of a² + p⁴
• Norm forms for arbitrary number fields as products of linear polynomials
• Universal torsors and values of quadratic polynomials represented by norms
• Universal torsors and values of quadratic polynomials represented by norms
• Higher dimensional analogues of Châtelet surfaces
• A Note On a Theorem of Heath-Brown and Skorobogatov
• Quadratic polynomials represented by norm forms
• On the distribution of reducible polynomials
• Rational solutions of certain equations involving norms
• The polynomial X²+Y⁴ captures its primes