Zero Shadows and Bulges of the Prime Zeta Function via GUE-Driven Scaling

Treat the prime zeta function P(s) as a nonlinearly filtered superposition of ζ(ks) across scales, then push the GUE paradigm for ζ-zeros (Rodgers, 2024) through this scaling map to derive predictions for micro- and mesoscopic organization of zeros of P(s). Specifically, investigate whether the lines Re s = 1/(2k) act as “resonant scaffolds” that create zero clusters and zero-poor corridors (“bulges”) for P(s), and quantify this with local zero-density and zero-spacing statistics. Combine this with a transfer-operator viewpoint for the map s ↦ ks and with explicit-formula-based heuristic calculations. Validate numerically by scanning zero sets of P(s) in boxes aligned with these scalings. This approach differs from prior work by adding a micro-local, mechanism-driven picture tied to ζ’s conjectural GUE statistics, exploiting the Möbius-weighted log-ζ expansion of P(s). If successful, it yields a principled map from conjectural ζ-zero statistics (GUE) to concrete, testable patterns in P(s), potentially explaining unexpected zero clustering or gaps and suggesting new zero-free strips/bulges not accessible by classical techniques. The impact includes illuminating the arithmetic content of P(s) and advancing understanding of how zeta’s random-matrix flavor propagates through arithmetic transforms.

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.