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

Technical Summary: Liouville Function, von Mangoldt Function and Norm Forms at Random Binary Forms

Problem Statement
This paper investigates the average behavior of arithmetic functions evaluated at the values of random binary forms of degree $d$. Specifically, it addresses three interconnected problems:

1. The Chowla Conjecture: Analyzing the cancellation of the Liouville function $\lambda(n)$ over values of binary forms.
2. The Bateman–Horn Conjecture: Establishing asymptotic formulas for the simultaneous prime values of tuples of binary forms, utilizing the von Mangoldt function $\Lambda(n)$.
3. The Hasse Principle: Determining the proportion of Châtelet varieties defined by norm forms that satisfy the rational Hasse principle.

While previous work by Browning, Sofos, and Teräväinen [5] established analogous results for random polynomials in one variable, this paper extends the framework to binary forms in two variables. The central challenge is to demonstrate that "almost all" binary forms (in a combinatorial sense) exhibit the expected statistical behavior predicted by these conjectures, despite the increased complexity of the underlying geometry and arithmetic.

Methodology
The paper employs a probabilistic and analytic number theory approach, centered on the concept of combinatorial cubes. A combinatorial cube $C$ is defined as a subset of coefficient vectors $(c_0, \dots, c_d) \in \mathbb{Z}^{d+1}$ where certain coefficients are fixed and others vary within a range $[-H, H]$. The goal is to show that for a set of coefficients $C$ of side length $H$, the "bad" forms (those failing the conjectures) constitute a negligible proportion, specifically $O((\log H)^{-A})$.

The core technical machinery involves:

• Equidistribution in Arithmetic Progressions: The paper generalizes a key tool from [5] (Theorem 2.1), which links the equidistribution of an arithmetic function in arithmetic progressions to its average behavior over random polynomials. This is adapted to binary forms by controlling sums of the form $\sum F(am^k n^{d-k} + bm^l n^{d-l} + g(m,n))$.
• Sieve Theory and Localized Counting: For the norm form problems, the author introduces a localized counting function $\hat{N}_c(x)$. This function approximates the global count of integer solutions to $N_K(x) = g_c(u)$ by restricting the search to a specific region $B$ and incorporating local densities (singular series) and archimedean densities.
• Approximation and Error Analysis: The proof strategy involves showing that the global counting function $N_c(x)$ is well-approximated by $\hat{N}_c(x)$ for almost all coefficients. This requires bounding the error term $|N_c(x) - \hat{N}_c(x)|$ and demonstrating that the localized function $\hat{N}_c(x)$ is rarely small (i.e., it is large enough to guarantee the existence of solutions).
• Geometric and Algebraic Constraints: The analysis distinguishes between "separable" forms and those with integer zeros or high content. Lemmas are provided to show that forms failing these "admissibility" conditions are rare. The proof also utilizes properties of the Dedekind zeta function and local densities over $p$-adic fields.

Key Contributions and Results

1. Averaged Chowla Conjecture for Binary Forms (Theorem 1.2):
   The paper proves that for almost all binary forms $g \in \mathbb{Z}[s, t]$ of degree $d$ with coefficients in a combinatorial cube, the sum of the Liouville function over the values of the form exhibits cancellation. Specifically, for $x \in [H^c, 2H^c]$,
   $$\sup_{x} \frac{1}{x^2} \left| \sum_{u,v \leq x} \lambda(g(u,v)) \right| \leq (\log H)^{-A}$$
   holds for all but a negligible fraction of forms.

2. Averaged Bateman–Horn Conjecture for Binary Forms (Theorem 1.3):
   The author establishes an asymptotic formula for the number of simultaneous prime values of an $r$-tuple of binary forms. They show that for almost all $r$-tuples, the sum of products of von Mangoldt functions satisfies:
   $$\sum_{m,n \leq x} \Lambda(g_1(m,n)) \cdots \Lambda(g_r(m,n)) \sim x^2 S_{g_1, \dots, g_r}(x)$$
   where $S$ is the product of local densities. This holds uniformly for $x$ in the specified range.

3. Rational Hasse Principle for Châtelet Varieties (Theorem 1.5):
   The paper proves that for a fixed norm form $N_K$ of degree $e$ and a varying binary form $g$ of degree $d$ (where $e|d$), the associated Châtelet variety $N_K(x) = g(u)$ satisfies the rational Hasse principle for 100% of coefficient vectors.
   $$\frac{\#S_{glob}(H)}{\#S_{loc}(H)} = 1 + O((\log H)^{-A})$$
   Here, $S_{loc}(H)$ represents forms locally solvable everywhere, and $S_{glob}(H)$ represents those with a rational point. The result implies that the Brauer–Manin obstruction is the only obstruction for almost all such varieties.

Significance and Claims
The paper claims to extend the "average" methodology of Browning, Sofos, and Teräväinen [5] from univariate polynomials to binary forms, a significant step given the higher dimensionality and different arithmetic properties of binary forms.

• Resolution of Colliot-Thélène's Conjecture (Average Case): The work provides an average version of a conjecture by Colliot-Thélène regarding the Hasse principle for Châtelet varieties. While previous results covered specific cases (e.g., linear polynomials or specific degrees), this paper proves the principle holds for 100% of norm form equations of the specified type, provided the degree of the norm divides the degree of the binary form.
• Quantitative Precision: The results are quantitative, providing explicit error terms of the order $(\log H)^{-A}$, which allows for a rigorous definition of "almost all" in the context of height-ordered binary forms.
• Methodological Advancement: The paper introduces a refined localized counting function and a detailed analysis of archimedean and non-archimedean densities to handle the norm form equations, overcoming the lack of homogeneity in the Châtelet variety equations.

The author explicitly states that their results prove the rational Hasse principle for 100% of the considered norm form equations, a stronger result than the integral Hasse principle previously established for polynomials in [5], which did not guarantee the rational case with probability 1 due to non-homogeneity. The paper does not claim to resolve the conjectures for every individual form, but rather establishes their validity in the statistical limit over the space of coefficients.