math.CV — Complex Variables
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 reveals four geometric mechanisms that distinguish non-trivial Riemann zeta zeros b = bn from generic values of b, including right-half-plane slow-zone clustering of P1 and opposinghemisphere alignment of the leading inner circles P3, P5, P7. Finally, we prove that removing all odd-even dyadic pairs leaves the multiples-of-4 subseries P∞ k=1 t4k(s) = t4(s)ζ(s), whose partial sums spiral around a target center Ctarget(b) that coincides with the origin (0, 0) if and only if b is a Riemann zeta zero.