k-bonacci Sequences and Zeckendorf Representation via Zeta Functions - A Unified Mathematical Framework

This paper systematically investigates the deep mathematical connections between the Riemann zeta function, k-bonacci sequences, and Zeckendorf representations. Through establishing an analytic framework for generalized Fibonacci zeta functions $\zeta_F^{(k)}(s)$, we reveal the intrinsic unity between recursive sequences, number-theoretic representations, and complex analysis. Core discoveries include: (1) Characteristic roots $r_k$ of k-bonacci sequences establish precise correspondence with zeta function negative integer values through Bernoulli numbers; (2) The no-k constraint naturally emerges in zeta functions through hierarchical compensation mechanisms, producing multi-dimensional negative information networks; (3) Uniqueness of Zeckendorf representations corresponds to the uniqueness of analytic continuation of zeta functions; (4) Time evolution operators realize computation-data duality through k-bonacci recursion and Fourier transforms of zeta functions; (5) Hierarchical structure of quantum decoherence precisely corresponds to the alternating sign pattern of $\zeta(-2n-1)$. This paper also establishes a complete physical application framework, predicting observable effects including quantum chaos spectra following k-bonacci distributions, high-energy physics energy gaps conforming to plastic number ratios, and fractal dimensions determined by multi-dimensional compensation networks. These results provide a unified mathematical foundation for understanding spacetime origins, quantum gravity, and information conservation.