Origin-Centered Möbius Circles and Dyadic Pairings of the Riemann Zeta Function on the Critical Line
We investigate the complex geometry of the normalized Dirichlet eta terms tn(s) = 1 1−2 1−s (−1)n−1 ns (s = a + bi). By analyzing the Möbius geometry of each term, we demonstrate that the base-2 resonant partial sum S2(s) = t1(s) + t2(s) forms an exact circle. Remarkably, this base circle is centered strictly at the origin (0, 0) if and only if Re(s) = 1 2 . We further establish a dyadic pairing theorem showing that every oddeven pair Pm(s) = tm(s) + t2m(s) inherits this origin-centered circular symmetry on the critical line. Analyzing the phase distribution on these concentric circles…
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