This paper systematically establishes a comprehensive framework for generalizing fast convergent summation formulas for the Riemann zeta function to complex orders and multivariate cases. By introducing functional equations and regularization techniques,we extend the fast-convergent formulas to the entire complex plane and provide global analytic expressions with correction terms. Furthermore, we conduct an in-depth investigation of the numerical stability issues of the Hurwitz zeta function for parameters 0 <a<1, proposing adaptive hybrid algorithms based on parameter transformations,series acceleration, and integral representations to enhance computational stability.For multiple zeta values (MZVs), we derive fast-convergent series expressions based on generating functions and integral representations, and design corresponding numerical algorithms. Additionally, we generalize this framework to Dirichlet L-functions,Dedekind zeta functions of algebraic number fields, L-functions of modular forms, Epstein zeta functions, and q-analogs and quantum zeta functions, revealing profound connections between fast-convergent formulas and the theory of modular forms. This work not only provides rigorous theoretical proofs and detailed algorithm designs but also validates the accuracy, efficiency, and universality of the proposed methods through extensive numerical experiments and applications in multiple fields including quantum field theory, statistical physics, string theory, and analytic number theory. Complete algorithm implementations and performance analyses are provided as appendices, offering powerful tools for scientific computing in related fields.
shifa liu (Wed,) studied this question.