Random Dedekind–Bateman–Horn Models with Local Symmetry Constraints

Generalize random Riemann zeta constructions by imposing local splitting constraints corresponding to primes represented by a given homogeneous polynomial f(a,b) or to Frobenius classes in number fields (Dedekind zeta context). Define a “random Dedekind–BH zeta” whose Euler factors follow constrained stochastic rules. Prove analytic continuation domains and study zero statistics: does the critical line survive? What deformations of pair correlation arise from local symmetry bias? Calibrate the model to reproduce Bateman–Horn main terms and then derive fluctuation/covariance predictions in short intervals, in the spirit of Rodgers (2024). This differs from typical random-zeta models that use Cramér-like pseudo-primes and often break at the critical line by encoding arithmetic structure directly into the randomness, creating a controlled sandbox to tune local data and read off global zero statistics and error-term fluctuations. This is a tractable laboratory for “what-if” scenarios hard to access analytically in the true setting. If certain local symmetries enlarge or shrink the domain of analytic continuation, that suggests which arithmetic features drive or suppress RH-like behavior and GUE statistics in genuine objects. The impact includes new heuristic laws for fluctuations in prime patterns represented by polynomials, concrete testable covariance formulas in short intervals, and a potential path to refine Bateman–Horn error terms by importing random-matrix/ratios heuristics into a locally realistic, globally analyzable model.

References:

1. Arithmetic Consequences of the GUE Conjecture for Zeta Zeros. B. Rodgers (2024). The Michigan mathematical journal.
2. Arithmetic Consequences of the GUE Conjecture for Zeta Zeros. B. Rodgers (2024). The Michigan mathematical journal.
3. Distribution of primes represented by polynomials and Multiple Dedekind zeta functions. I. Horozov, Nickola Horozov, Zouberou Sayibou (2022).