Primorial Phase-Locking Tests for Deviations from GUE in Zeta Zeros

Define “phase-locking heights” T where the phases t log p (mod 2π) for primes p dividing a primorial P# are unusually aligned. Around these T, measure deviations in local zero spacing, one-level density, and statistics of (ζ’/ζ)(1/2+it) from GUE predictions. Calibrate against Rodgers’ arithmetic consequences of GUE (2024) and ratio-conjecture style predictions for correlations of ζ and its derivative. Complement with time-series tests for quasi-periodicity and autocorrelation noted in Milgram–Hughes (2024). The outcome is a suite of pre-registered numerical experiments with statistical power to accept/reject “primorial anomaly” effects. This differs from prior qualitative claims by formalizing quantitative tests with a clearly specified null hypothesis and power analysis, connecting to explicit-formula variants to design phase functionals sensitive to primorial alignment. Either confirming or rejecting primorial anomalies informs understanding of arithmetic phase alignment effects on ζ-zero statistics. The impact is establishing a rigorous, reproducible framework for adjudicating high-profile anomaly claims and potentially revealing subtle departures from random-matrix universality.

References:

1. Arithmetic Consequences of the GUE Conjecture for Zeta Zeros. B. Rodgers (2024). The Michigan mathematical journal.
2. On a generalized moment integral containing Riemann’s zeta function: Analysis and experiment. Michael Milgram, Roy Hughes (2024). Modern Mathematical Methods.