Certified Arbitrary-Precision Evaluation of a Family of Generalized Multiple Zeta Functions

Abstract

We describe a certified arbitrary-precision framework for evaluating a family of generalized multiple zeta functions. The family includes strict and weak-star chain sums, ordinary and colored multiple zeta values, affine-base and polynomial-base variants, and composite levels containing several affine or polynomial letters with complex coefficients. The numerical strategy combines finite-prefix recurrences with two complementary analytic-tail mechanisms: recursive Euler-Maclaurin expansion of one-variable tails and direct absolute tail majorants. The Euler-Maclaurin branch is fast when the relevant suffix expansions are regular, while the direct-tail branch gives robust certificates for multi-letter, weak-star, complex-coefficient, and branch-sensitive inputs. A computation is called certified only when its reported radius is obtained from a proved analytic bound for the omitted infinite tail. Strict-disk colored sums and boundary-color cases with summable absolute majorants are therefore within the certified scope; conditionally convergent colored cases whose convergence relies only on non-one unit-modulus oscillation are kept separate and reported as explicitly non-certified diagnostic outputs unless an independent analytic remainder bound is available.