Worse than square-root cancellation in Bateman-Horn's conjecture

Abstract: We prove asymptotics for the average error term in Bateman-Horn's conjecture in the exponential range.

• The paper establishes that the average second moment of the Bateman–Horn error term grows as x log H, proving fluctuations exceed square-root cancellation for polynomials of degree at least two.
• It leverages Hooley’s neutraliser method combined with Brun’s sieve to dissect diagonal and off-diagonal contributions, providing sharper bounds than earlier approaches.
• The findings demonstrate that conventional square-root cancellation fails, guiding computational studies and setting new benchmarks for estimating error terms in prime-producing polynomial sequences.

Worse than Square-Root Cancellation in Bateman–Horn's Conjecture

Overview and Motivation

This paper establishes refined asymptotic behavior for the average second moment of the Bateman–Horn error term, particularly in the regime where $x$ grows exponentially in $\log H$, where $H$ bounds the coefficients of degree-$d$ integral polynomials considered. The result confirms that, in this regime, the fluctuation of the error surpasses the conventional square-root barrier, elucidating that square-root cancellation does not hold after averaging for polynomials of degree at least two. The work leverages Hooley's neutraliser method combined with Brun's sieve, substantially improving previous bounds (2604.02287).

Theoretical Framework

Let $P(t)\in\mathbb{Z}[t]$ denote an irreducible polynomial of degree $d\geq 2$ and positive leading coefficient. The Bateman–Horn conjecture predicts the asymptotic number of prime values taken by $P(n)$ for $n\le x$, formalized via the function

$$\psi_P(x) = \sum_{1 \le n \le x,\; P(n)>0} \Lambda(P(n)),$$

where $\Lambda$ is the von Mangoldt function. The conjecture expects

$\log H$0

where $\log H$1 is the Bateman–Horn constant, encoded as an infinite product involving local root counts modulo primes.

While the conjecture is established for linear polynomials ($\log H$2) by Dirichlet’s theorem, all higher degree cases remain open. Recent research has moved toward average-case analysis, considering families of polynomials with bounded coefficients, denoted $\log H$3.

A central and unresolved aspect is the size of the error term $\log H$4, and whether square-root cancellation, i.e., a bound of $\log H$5 (after averaging), can hold in any meaningful regime.

Main Results

The principal achievement is an asymptotic for the second moment:

$\log H$6

valid for $\log H$7 in the exponential regime, $\log H$8, for any fixed $\log H$9.

Key Consequences

• Violation of Square-Root Cancellation: For $H$0, the fluctuations average to $H$1, which is stronger than the square-root barrier that would suggest $H$2.
• Improvement over Prior Bounds: Previous work (Skorobogatov–Sofos) attained only an $H$3 upper bound for the mean-square, under much more restrictive truncation of $H$4. Here, both the truncation and the error are sharper.
• Comparative Bounds for Higher Moments: For odd moments and large $H$5, the result outperforms upper bounds from Kravitz–Woo–Xu for a broad range of parameters.

Methodology

Use of Hooley's Neutralisers

Traditional approaches (e.g., Skorobogatov–Sofos) employed severe truncation of the Bateman–Horn constant and circle method arguments, which inherently limited their range of validity and weakened the error terms. This paper’s methodological advance is integrating Hooley’s neutralisers—analytic devices that allow smoother treatment of truncation—enabling control at much larger $H$6 in the truncated product defining $H$7.

Sieve and Mean Square Expansion

The mean square is expanded and dissected into diagonal and off-diagonal (“cross”) terms, both of which are evaluated precisely. The diagonal terms reduce via analytic number theory to local computations, substantiated by explicit treatment of sums involving the von Mangoldt function. Off-diagonal terms are analyzed using a blend of combinatorial sieving, careful application of Bombieri–Vinogradov–type bounds, and the refinement given by the neutralisers.

The analysis is highly sensitive to the choice of truncation points and relies on nontrivial bounds for sums over value distributions and the application of deep results in multiplicative number theory.

Implications

Theoretical Implications

This paper demonstrates explicitly that any approach to unconditional results for the Bateman–Horn conjecture cannot simply assume square-root cancellation after averaging, even for generic families of polynomials with large coefficients unless $H$8 is polynomial in $H$9. This clarifies the limitations inherent in analytic number theory techniques for higher-degree polynomial prime values.

The findings also yield refined tools for unconditional probabilistic models of prime-producing polynomials, feeding into the framework for understanding phenomena such as the polynomial Chowla and Hasse principles on average.

Practical Perspectives

The precise form of the error term is critical for computational investigations seeking to test or approximate the distribution of primes of the form $d$0. Knowing that the mean-square deviation is of order $d$1 prevents overoptimistic expectations regarding the variance of empirical counts for moderate values of $d$2 and $d$3.

Future Directions

A compelling challenge is to extend these asymptotics further into the regime $d$4 by possibly developing alternative averaging or sieving techniques, as the current machinery is insufficient for such “short intervals.” The possibility of extending the analysis to all higher moments (not just the second) or to more general families of polynomials (e.g., reducible ones or over number fields) remains open. Additionally, interactions with the distribution of zeros of associated $d$5-functions and Random Matrix Theory heuristics warrant further exploration.

Conclusion

This work provides strong evidence that the error term in the Bateman–Horn conjecture, after averaging over polynomial families in the prescribed ranges, typically exceeds what square-root cancellation would predict. The novel integration of Hooley’s neutralisers into this analytic context not only strengthens the theoretical understanding of error terms in polynomial prime value counts but also advances practical techniques for future research on the distribution of prime values of polynomials (2604.02287).