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Explore the mathematical landscape of zeta functions in this 24-minute conference talk that bridges classical number theory with modern physics applications. Discover how zeta functions, including the Riemann zeta and multiple zeta values (MZVs), along with harmonic numbers and polylogarithms, form the fundamental language for solving complex problems spanning analytic number theory, combinatorics, and multiscale Feynman integrals. Learn about a comprehensive family of mathematical symbols—AlternatingHarmonicNumber, HyperHarmonicNumber, MultipleHarmonicNumber, MultipleZeta, HarmonicPolyLog, MultiplePolyLog, and GeneralizedPolyLog (GPL)—that provide exact evaluation capabilities, high-precision numerics, shuffle/stuffle-aware simplification, and branch-aware transformations with deep integration into computational systems. Master techniques to collapse high-weight Euler sums into canonical MZV combinations, normalize mixed products of harmonic numbers, logs, and polylogs for downstream computations, solve regular and perturbative Fuchsian systems directly using GPLs, and verify complex mathematical results through high-precision numerical checks. Begin with an accessible historical overview of zeta functions before diving into the unified computational interface that connects Euler sums, harmonic number transformations, iterated integrals, and Fuchsian systems in a coherent mathematical framework.
