Bateman-Horn, polynomial Chowla and the Hasse principle with probability 1

Technical Summary: Bateman–Horn, Polynomial Chowla and the Hasse Principle with Probability 1

Problem Statement
The paper addresses three fundamental problems in analytic number theory concerning the behavior of arithmetic functions and Diophantine equations when evaluated at polynomials $f \in \mathbb{Z}[t]$ of degree $d \geq 1$. Specifically, the authors investigate the average behavior of these objects when the coefficients of the polynomials are chosen randomly from a combinatorial cube (a box in coefficient space).

1. The Bateman–Horn Conjecture: This conjecture predicts the asymptotic density of prime values taken by an irreducible polynomial $f(t)$. While known for linear polynomials, it remains out of reach for fixed polynomials of degree $d \geq 2$. The paper seeks to establish this conjecture for "almost all" polynomials of a given degree.
2. The Polynomial Chowla Conjecture: This conjecture predicts that the Liouville function $\lambda(f(n))$ exhibits cancellation (sums to $o(x)$) for any polynomial $f$ that is not of the form $c g(t)^2$. Like the Bateman–Horn conjecture, this is open for irreducible polynomials of degree $\geq 2$.
3. The Integral Hasse Principle for Norm Forms: The paper studies the solubility of the equation $N_K(\mathbf{x}) = f(t)$ over the integers, where $N_K$ is the norm form of a number field $K/\mathbb{Q}$ and $f(t)$ is a polynomial. The integral Hasse principle asserts that local solubility (over $\mathbb{R}$ and $\mathbb{Z}_p$ for all $p$) implies global solubility over $\mathbb{Z}$. The authors aim to determine if this principle holds for a typical polynomial $f$.

Methodology
The core technical innovation is a reduction of the problem of averaging arithmetic functions over random polynomials to the problem of equidistribution in short intervals and arithmetic progressions.

• The Key Technical Tool (Theorem 2.2): The authors prove that if an arithmetic function $F(n)$ is well-distributed in arithmetic progressions (specifically, if its sums over intervals $I$ in residue classes $u \pmod q$ are small for "most" moduli $q$), then the sum of $F(f(n))$ over random polynomials $f$ exhibits significant cancellation. This result relies on a second-moment method and careful analysis of the anatomy of integers (divisors, smooth numbers) to handle the correlation between polynomial values.
• Equidistribution of $\Lambda$ and $\lambda$: To apply the main tool, the authors establish that the von Mangoldt function $\Lambda$ and the Liouville function $\lambda$ satisfy the required equidistribution properties in short intervals and progressions, excluding a sparse set of "bad" moduli. This utilizes zero-density estimates for Dirichlet $L$-functions and the zero-free regions of $L(s, \chi)$.
• Norm Forms and Localized Counting: For the Hasse principle problem, the authors construct a "localized counting function" $\hat{N}_c(x)$ that approximates the number of integer solutions to $N_K(\mathbf{x}) = f(t)$. This function combines a local density factor (derived from the Dedekind zeta function and local solubility) with an archimedean density factor (a singular integral). They prove that the actual counting function $N_c(x)$ is well-approximated by $\hat{N}_c(x)$ on average and that $\hat{N}_c(x)$ is rarely small.

Key Contributions and Results

1. Bateman–Horn with Probability 1 (Theorem 1.2 & 1.4):

   • The authors prove that for almost all polynomials $f$ of degree $d$ with coefficients in a combinatorial cube of side length $H$ (where the constant term is not fixed to 0), the Bateman–Horn asymptotic holds for sums of length $x \asymp H^c$ with $c < 5/(31d)$.
   • This improves upon previous work (e.g., Skorobogatov and Sofos) by allowing the summation range $x$ to be polynomial in $H$ rather than logarithmic, and by requiring averaging over only two coefficients (dimension of the cube $\geq 2$) rather than all coefficients.
   • Corollary 1.3: A typical admissible polynomial takes at least $H^{5/(31d) - \epsilon}$ prime values.

2. Polynomial Chowla with Probability 1 (Theorem 1.5):

   • The paper establishes that for almost all polynomials $f$ of degree $d$, the sum $\sum_{n \leq x} \lambda(f(n))$ is bounded by $x (\log H)^{-A}$ for $x \in [H^c, 2H^c]$. This confirms the polynomial Chowla conjecture for almost all polynomials, improving on qualitative results by Teräväinen.

3. Integral Hasse Principle for Generalized Châtelet Surfaces (Theorem 1.6):

   • For the equation $N_K(\mathbf{x}) = f(t)$, the authors prove that if $f$ is "admissible" (leading coefficient positive and locally soluble everywhere), then the equation has an integer solution with probability 1.
   • The error term in the proportion of solvable polynomials saves an arbitrary power of $\log H$.
   • Corollary 1.8: For 100% of admissible polynomials, the integer solutions are Zariski dense (not contained in a finite union of low-degree curves).
   • This result generalizes previous work by Derenthal and Wei (who handled specific cases like $K=\mathbb{Q}(i)$ and quadratic $f$) to arbitrary number fields and arbitrary degree polynomials, without relying on Schinzel's Hypothesis.

Significance and Claims
The paper claims to resolve the average behavior of these three major conjectures/problems simultaneously.

• Strength of Results: The authors emphasize that their results hold for "almost all" polynomials (with an exceptional set of size $O(\#C / (\log H)^A)$), which is a stronger statement than proving results for a positive proportion of polynomials.
• Independence from Unproven Hypotheses: Unlike many results in this field that rely on Schinzel's Hypothesis or the Generalized Riemann Hypothesis (GRH), these results are unconditional. The range of $x$ achieved is comparable to what GRH would provide (up to the constant $c$), but the proof is unconditional.
• Methodological Advance: The reduction to equidistribution in progressions (Theorem 2.2) is presented as a versatile tool that decouples the difficulty of the specific arithmetic function (whether it is $\Lambda$, $\lambda$, or a norm form representation function) from the difficulty of the polynomial structure, provided the function is well-distributed.
• Hasse Principle: The paper highlights that for the integral Hasse principle, the Brauer–Manin obstruction is the only obstruction for 100% of admissible polynomials, a result that extends beyond the scope of previous work which was limited to specific fields or required Schinzel's Hypothesis.

The authors note that the size of the exceptional set and the exponent $c$ are limited by current knowledge of the Siegel–Walfisz theorem and zero-density estimates, respectively, and that further improvements in these areas would directly improve the quantitative bounds in their theorems.