No Evidence for Detectable Arithmetic Frequencies in Long-Range Deviations of Riemann Zero Statistics from Random Matrix Theory

This preprint investigates whether arithmetic structure associated with prime numbers can be detected in the long-range deviations of the Δ₃ statistic from Random Matrix Theory predictions for the nontrivial zeros of the Riemann zeta function. Using 60 evaluation points derived from zeros at height T ∼ 10⁸–10⁹, we compare five statistical models via residual sum of squares and the Bayesian Information Criterion. To account for correlations in the data, we apply block bootstrap resampling. As a control against overfitting, we include a model with freely optimized frequencies (Model 1b). While a statistically significant periodic structure exists in the residuals, none of the arithmetic models (based on frequencies ln p, ln(pq), or their combinations) provides a competitive explanation. The free-frequency model significantly outperforms all arithmetic constructions (ΔBIC = 21.86). Even the pure Random Matrix Theory baseline attains a better BIC value than the arithmetic models. These results indicate that, at the examined height, the observed long-range deviations of Δ₃(L) cannot be attributed to the simplest arithmetic frequencies in an additive form. The work provides a rigorous statistical test of concrete arithmetic hypotheses and outlines directions for future research at greater heights.