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Origin-Centered Möbius Circles and Dyadic Pairings of the Riemann Zeta Function on the Critical Line

A mix of human-written and AI-generated textHuman understanding: all partsmath.CV — Complex Variables

Contributed by Ryan Hackbarth ↗

Version 1 / Oct 23, 2026 / CC BY 4.0

Abstract

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.

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References

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  3. T. Needham, Visual Complex Analysis , Oxford University Press, 1997.
  4. J. B. Garnett, Bounded Analytic Functions , Revised 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, 2007.
  5. H. Bohr and R. Courant, ``Neue Anwendungen der Theorie der Diophantischen Approximationen auf die Riemannsche Zetafunktion,'' Journal für die reine und angewandte Mathematik , vol. 144, pp. 249--274, 1914.
  6. T. M. Apostol, Introduction to Analytic Number Theory , Undergraduate Texts in Mathematics, Springer-Verlag, 1976.

Version history

  1. v1Submitted by Ryan HackbarthInitial depositCurrentOct 22, 2026