Based on the established theory for the complex-order Riemann zeta function ζ(s) (s ∈ C), encompassing closed-form expressions, Γ-rational approximations, and complex fractional differential equations, this paper systematically deepens and extends the related mathematical framework, proposing and proving several new theorems. In theoretical deepening, we rigorously establish an explicit formula connecting the zero distribution of ζ(k)(s) and higher-order moments of prime number distribution (Theorem 2.1), derive and verify a precise symmetric form of the functional equation for higher derivatives (Theorems 2.2 & 2.3), and preliminarily construct the L-function theory corresponding to complex-weight modular forms (Theorem 2.4). In zero-point research, we strictly prove the local quadratic convergence of a zero-finding iteration algorithm based on an integral equation (Theorem 3.1), deepen the constraint relations on zeros derived from complex fractional differential equations providing their specific asymptotic expansions (Theorem 3.2), and rigorously transform conjectures on zero statistics into a spectral distribution problem for compact integral operators (Theorem 3.3). For multidimensional extensions, we completely prove a universal contour integral representation theorem for complex-order multiple zeta functions of arbitrary multiplicity k (Theorem 4.1) with a rigorous proof of convergence (Theorem 4.2), and provide an upper bound estimate for their irrationality measure at complex algebraic points (Theorem 4.3). Furthermore, this paper designs and verifies core algorithms and formulas for parallel computation, adaptive precision control, and cross-disciplinary application models. All conclusions are supported by rigorous proofs or detailed derivation frameworks, and theoretical verifiability is ensured through designed numerical experiment schemes.
shifa liu (Wed,) studied this question.