Higher Derivative Theory of the Alternating Dirichlet Eta Function: Zero Distribution, Structural Properties, and Computational Simplification

This paper systematically investigates the analytic properties, zero distribution, and computational theory of the higher-order derivatives η(k)(s) of the Dirichlet eta function. Based on the contour integral representation, we establish precise expressions for η(k)(s), revealing its intrinsic connection with the zeros of the Riemann zeta function. Through in-depth analysis of the symmetry, cyclic recurrence relations, and sparsity structure of η(k)(s), we derive its strict symmetric properties on the critical line, obtain exact zero-counting formulas, and prove the validity of the derivative version of the Generalized Riemann Hypothesis under specific conditions. In particular, we systematically utilize symmetry reduction, cyclic recurrence, and sparsity patterns to develop a series of efficient algorithms that reduce the computational complexity from exponential to polynomial levels, providing a complete mathematical framework for large-scale numerical computation and theoretical analysis. This paper also establishes a generalized theoretical framework from one-dimensional to multi-dimensional cases, laying the foundation for the study of complex-order multiple L-functions.