Based on the core framework proposed in Liu’s paper 1 (“Rapidly Convergent Summation Formulas for the Riemann Zeta Function of Complex Order and Multivariate Generalizations with Extensions”), this paper systematically plans and deeply deduces ten expansion directions of this framework in the frontier fields of future mathematics and computational science. The ten directions are: (1) Generalization to Dirichlet L-functions, (2) Generalization to automorphic form L-functions, (3) Extension to graph zeta functions, (4) Extension to fractal spectral zeta functions, (5) Fast computation of multiple zeta values of arbitrary depth, (6) High-performance computing and specialized hardware acceleration, (7) Cryptographic applications, (8) Machine learning activation functions, (9) Experimental mathematics and high-precision computation, and (10) Stability analysis in regions with large imaginary parts. For each task, we not only construct detailed theoretical generalization paths and rigorous mathematical proof frameworks but also design feasible algorithmic implementation schemes, exploring their potential applications in interdisciplinary fields such as cryptography, machine learning, and physical sciences. This work aims to extend the rapidly convergent method from the classical zeta function to more general L-functions, graph and fractal zeta functions, arbitrarily depth multiple zeta values, and to integrate high-performance computing (GPU/FPGA),quantum algorithms, and high-precision experimental mathematics. Ultimately, we aim to construct a unified, efficient, and high-precision platform for special function computation and theoretical analysis, opening new avenues for the deep integration of number theory, mathematical physics, and computational science.
shifa liu (Thu,) studied this question.