Dirichlet Eta Function of Prime Distributions, Fractal Geometry, and Gravitational Wave Dynamics

The Dirichlet eta function, often referred to as the "alternating zeta function," is a crucial intersection of analytic number theory, fractal geometry, and modern physics. While it shares deep connections with the Riemann zeta function, its alternating series structure and unique analytic properties uncover hidden symmetries in prime distributions, recursive dynamics, and spacetime perturbations. This research investigates the interconnectedness of number theory, fractal geometry, and gravitational wave physics through the lens of the Dirichlet eta function. We introduce a novel prime product representation, highlighting the role of the even prime (2) as a path selector that governs scale-invariant dynamics across diverse scales. Key Findings and Contributions: Parity-Asymmetric Prime Distribution: We demonstrate a modulation that incorporates prime correlatives into exclusive corrections within the Dirichlet eta function's product formula, revealing an inherent asymmetry in prime distribution. Fractal Clustering of Zeros: We observe fractal clustering in the zeros of the eta function, with spacing characterized by Fibonacci numbers (Tn+2 = Tn+1 + Tn, T1 = T2 = 1), indicating a structure mediated by metallic means, including the silver and plastic numbers. Gravitational Wave Echo Spectra: We present new predictions for gravitational wave echo spectra from exotic compact objects (ECOs), derived from the eta function's structural properties. Employing Pöschl-Teller potential analogs, we show that scale intervals correspond to length-dependent signals, proportional to log(2)^n. Metallic Mean Field Theory (MMFT) Extension: We extend the MMFT framework to explore phase transitions in ECOs, establishing a link between prime distribution and the structural properties of primes. Dyadic Zeta Functions (DZF) Expansion: We expand the scope of the DZF through prime gap thermodynamics and AdS/CFT holography, providing new perspectives on the structure of primes and the universe. Quantification of Chebyshev bias through log-periodic single-frequency oscillations. This work establishes the Dirichlet eta function as a fundamental geometric relation, connecting quantum arithmetic, fractal cosmology, and relativistic gravity.