Abstract

Starting from a compact planar set of measure (1-ln2), previously shown to encode the values ζ(k) for integers k≥2, we introduce a family of real parameters R_k whose associated boundary points x_k∈1,2 satisfy a transcendental equation involving the branch W_(-1) of the Lambert W function. We show that the transition from the divergent harmonic series ζ(1) to its analytically continued finite part admits a geometric interpretation governed by the unique inflexion point of W-_(1), which maps the boundary point x_1=1 of a rectangular compact of area R_1=2. Extending the construction to s=0 yields R_0=γ+ln2+1, consistent with ζ0=-1/2. Using the functional equation, we further show that the same geometric mechanism extends coherently to all negative integers, with the trivial values ζ(-n) appearing as rational increments in the extended sequence {R_k}_k∈Z. Altogether, these results highlight the structural role of the Lambert W_(-1) function in the analytic behaviour of ζ(s) and provide a unified geometric interpretation of its continuation at all integer arguments.

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