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We derive the structural relations between nested harmonic sums and thecorresponding Mellin transforms of Nielsen integrals and harmonicpolylogarithms at weight {\sf w = 6}. They emerge in the calculations ofmassless single--scale quantities in QED and QCD, such as anomalous dimensionsand Wilson coefficients, to 3-- and 4--loop order. We consider the set of themultiple harmonic sums at weight six without index $\{-1\}$. This restrictionis sufficient for all known physical cases. The structural relations supplementthe algebraic relations, due to the shuffle product between harmonic sums,studied earlier. The original amount of 486 possible harmonic sums contributingat weight {\sf w = 6} reduces to 99 sums with no index $\{-1\}$. Algebraic andstructural relations lead to a further reduction to 20 basic functions. Thesefunctions supplement the set of 15 basic functions up to weight {\sf w = 5}derived formerly. We line out an algorithm to obtain the analyticrepresentation of the basic sums in the complex plane.

Structural Relations of Harmonic Sums and Mellin Transforms at Weight w=6