This paper presents a systematic extension and deepening of the fast convergence summation method for alternating Riemann zeta functions proposed by Liu1. From theoretical and applied perspectives, we conduct comprehensive expansions and enhancements. On the theoretical side, we deeply explore mathematical properties such as symmetry, periodic structure, and sparsity, establishing a complete framework of symmetry reduction theory, periodic recurrence relations, and adaptive sparse algorithms. This reduces the computational complexity from O(K2) to O(√K logK) while maintaining super-exponential convergence. On the application side, we systematically extend the method to six important directions: (1) general Artin L-functions and automorphic forms; (2) fast algorithms for modular form related functions; (3) GPU/FPGA acceleration and quantum algorithm exploration; (4) applications in cryptography and machine learning; (5) development of an open-source high-performance mathematical library; and (6) combinatorial and number-theoretic significance of alternating multiple zeta values. This paper transforms all original conjectures and speculations into rigorous theorems, providing complete mathematical proofs, algorithm designs, and numerical verifications, demonstrating the powerful potential of this method both theoretically and computationally.
shifa liu (Wed,) studied this question.