Estimating the tail of the singular product for the Hardy Littlewood and Bateman Horn conjectures

Abstract: This paper investigates the asymptotic behavior of the tail of the singular product arising in the Hardy Littlewood and Bateman Horn conjectures for one dimensional systems of polynomials. A universal estimate is proved, showing that the contribution of large primes decays like the reciprocal of the logarithm, regardless of the structure of the system. For linear systems (trivial Galois group) superfast convergence is obtained. For nonlinear systems a coefficient is defined that is expressed via the average over the Galois group; in the abelian case and under the Riemann Hypothesis for Dirichlet L functions a more precise error estimate is obtained. Mixed systems containing both linear and nonlinear polynomials are also considered. Numerical experiments, presented as summary tables, confirm the theoretical conclusions. The results provide a rigorous theoretical foundation for computing singular series and refine the Bateman Horn formula.

• The paper establishes a universal upper bound on the tail decay rate of singular products in polynomial systems, showing a logarithmic decay of O(1/log q).
• The paper demonstrates that convergence rates depend on the Galois group, with linear systems converging superfast and nonlinear cases requiring GRH for faster decay.
• The paper provides precise error estimates and numerical validation, offering practical tools for truncating computations in prime value studies.

Asymptotics of the Singular Product Tail for the Hardy-Littlewood and Bateman-Horn Conjectures

Problem Statement and Context

This paper rigorously analyzes the asymptotic behavior of the "tail" of the singular product appearing in the Hardy-Littlewood and Bateman-Horn conjectures for one-dimensional systems of polynomials (2606.28832). Both conjectures link the density of prime-producing values of polynomial systems to a singular product—an infinite Euler-like product over primes capturing local arithmetical obstructions. Accurate estimation and truncation of this product are crucial for explicit computations, especially since practical calculations must truncate these infinite products at finite bounds.

The central problem is to quantify how quickly the tail—i.e., the contribution of large primes, exceeding a given threshold $q$—decays as $q \to \infty$. The work pays particular attention to differences arising from the Galois group structure of the associated splitting field and examines linear, nonlinear, and mixed polynomial systems.

Main Results

Universal Estimate and Galois Group Dependence

Theorem 3.1 establishes a universal upper bound for any one-dimensional polynomial system: the logarithm of the tail decays at a rate $O(1/\log q)$, independent of the specific polynomial system. This result confirms that negligible truncation errors are achieved for large $q$ in all practical contexts, but more refined asymptotics arise upon specialization.

The author introduces a coefficient $C(F)$ in the general tail asymptotic for nonlinear and mixed systems, expressible in terms of an average over the Galois group action via fixed points of Frobenius elements. For abelian Galois groups, the situation simplifies: $C(F) = k - m$, where $k$ is the number of input polynomials and $m$ is the count of distinct irreducible factors. This coefficient determines whether the singular product converges ($C(F) = 0$), diverges to infinity ($C(F) > 0$), or diverges to zero ($q \to \infty$0).

Explicit Asymptotics by System Type

Linear systems (i.e., all polynomials irreducible linear forms over $q \to \infty$1) realize the trivial Galois group. Here, the tail converges superfast: $q \to \infty$2 for $q \to \infty$3 polynomials. This rate notably outpaces all nonlinear or mixed cases and does not require unproven hypotheses such as GRH. All cases in classical Hardy-Littlewood settings (e.g., k-tuple conjectures, Goldbach-type configurations) exhibit this rapid convergence.

For nonlinear systems with abelian Galois groups and $q \to \infty$4 (i.e., convergent product), the author establishes:

• Without GRH: $q \to \infty$5, indicating slower convergence than the linear case.
• With GRH: $q \to \infty$6, delivering a polynomial rate of decay in $q \to \infty$7 for any $q \to \infty$8.

This acceleration under GRH leverages classical effective versions of the Chebotarev density theorem and analytic properties of Dirichlet $q \to \infty$9-functions.

For mixed systems (containing both linear and nonlinear polynomials), the tail asymptotics and convergence type follow from the general abelian formula, reducing to the evaluation of $O(1/\log q)$0 as above.

Singular Product Behavior and Numerical Verification

The partial singular product $O(1/\log q)$1, formed by multiplying local factors up to prime $O(1/\log q)$2, converges to its limiting value at a rate determined by the tail $O(1/\log q)$3. For abelian cases (with $O(1/\log q)$4), $O(1/\log q)$5. Under GRH, convergence accelerates to $O(1/\log q)$6. For linear systems, $O(1/\log q)$7 achieves $O(1/\log q)$8 at rate $O(1/\log q)$9, independent of GRH.

The author provides numerical evidence consistent with these results, showing, for instance, that errors for nonlinear (abelian group) systems are $q$0 at $q$1 under GRH, while linear systems achieve an error $q$2 at $q$3 without GRH.

Divergent Products and Limiting Cases

When $q$4, the singular product fails to converge—diverging to zero or infinity depending on the sign. Such degenerate systems fall outside the scope of the classical conjectures, reinforcing the strict admissibility conditions inherent in Bateman-Horn-type statements.

Theoretical and Practical Implications

This work supplies a rigorous justification for the truncation of the singular series in explicit computations concerning prime values of polynomials, providing practitioners with precise error controls based on easily computable group-theoretic invariants. The results enable reliable error assessment in experiments or attempts to verify conjectures numerically for Hardy-Littlewood or Bateman-Horn patterns.

By demonstrating that fast convergence in the linear/trivial group case is unconditional, while nonlinear systems benefit from GRH, the paper clarifies the critical role played by the arithmetic and Galois-theoretic structure of the system. The explicitness of the coefficient $q$5 also provides diagnostic information for the permissibility and convergence of the singular product in general mixed systems.

On a theoretical level, the paper refines our understanding of the transition from slow (logarithmic) to fast (power-law) convergence rates, specifically elucidating where GRH exerts significant quantitative influence over analytic number-theoretic estimates. This deepens the link between algebraic properties (Galois group, irreducibility) and analytic behavior (rate of convergence), contributing to the broader program of "explicit" analytic number theory.

Future Developments

The author outlines several avenues for further investigation:

• Development of asymptotics for systems with non-abelian Galois groups, likely requiring progress in Artin $q$6-function theory and potential resolution of the GRH for these cases.
• Systematic classification of convergence types for systems with dependent Galois groups or intricate factorization patterns.
• Extension to multivariate cases and examination of the impact of cohomological invariants in higher dimensions, building on previous work on geometric and trace function methods.

Additionally, improvements in effective error bounds could further tighten computational applications, especially if hypothetical Siegel zeros are ruled out or better understood.

Conclusion

The paper delivers a detailed asymptotic analysis of the tail of the singular product in the Hardy-Littlewood and Bateman-Horn conjectures for one-dimensional polynomial systems. The established universal bound, explicit formulae valid in the abelian and linear cases, and the delineation of the impact of GRH provide sharp tools for both theoretical investigations and computational experiments concerning prime values of polynomials. The central insight is that the convergence behavior is rigidly determined by the arithmetic nature of the system, particularly the Galois group and its action on roots, thus setting a precise standard for admissibility and error control in related analytic number-theoretic problems (2606.28832).