HEXAGONbeta

Origin-Centered Mobius Circles, Bernoulli Lemniscates, and Beta–Gamma 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 Variablesmath.HO — History and Overviewmath.NT — Number Theory

Contributed by Ryan Hackbarth ↗

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

We investigate the complex geometry of the normalized Dirichlet eta terms tn(s)=11−21−s(−1)n−1nst_n(s) = \frac{1}{1 - 2^{1-s}} \frac{(-1)^{n-1}}{n^s} for s=a+bis = a + bi. Whereas individual terms on the critical line Re⁡(s)=12\operatorname{Re}(s) = \frac{1}{2} undergo a nearly sixfold modulus oscillation driven by the Euler prefactor (1−21−s)−1(1 - 2^{1-s})^{-1}, the two-term partial sum S2(s)=t1(s)+t2(s)S_2(s) = t_1(s) + t_2(s) traces an exact Mobius circle for every fixed a≠1a \neq 1 that is centered at the origin (0,0)(0,0) if and only if Re⁡(s)=12\operatorname{Re}(s) = \frac{1}{2}, where it reduces to a scaled Blaschke factor 12B(z)\sqrt{\frac{1}{2}} B(z) of constant radius 12\sqrt{\frac{1}{2}}. Factoring m−sm^{-s} across odd-even dyadic pairs Pm(s)=tm(s)+t2m(s)=m−sS2(s)P_m(s) = t_m(s) + t_{2m}(s) = m^{-s} S_2(s) eliminates prefactor amplitude oscillations entirely, collapsing precessing annular rosettes into concentric origin-centered circles on the critical line with strictly descending radii Rm=12mR_m = \sqrt{\frac{1}{2m}}. Removing all odd-even pairs leaves the complementary multiples-of-44 terms t4k(s)=k−st4(s)t_{4k}(s) = k^{-s} t_4(s): we prove that on the critical line the anchor term t4(12+ib)t_4(\frac{1}{2} + ib) traces an exact Lemniscate of Bernoulli with foci at (0,0)(0,0) and (1,0)(1,0) (uniquely for Re⁡(s)=12\operatorname{Re}(s) = \frac{1}{2}), pure powers of 22 (k=2rk = 2^r) trace closed periodic multi-loop rational curves, and non-powers of 22 trace precessing lemniscate rosettes, while the dyadic differences D4m(s)=t4m(s)−t8m(s)=−4−sPm(s)D_{4m}(s) = t_{4m}(s) - t_{8m}(s) = -4^{-s} P_m(s) collapse both families into exact origin-centered circles, partitioning the sequence {tn(s)}n=1∞\{t_n(s)\}_{n=1}^{\infty} termwise and decomposing ζ(s)\zeta(s) via analytic continuation. Finally, we establish a pointwise velocity--modulus duality ∣dt4dθ∣=∣t2(12+ib)∣=n2∣tn(12+ib)∣\left|\frac{dt_4}{d\theta}\right| = \left|t_2\left(\frac{1}{2} + ib\right)\right| = \sqrt{\frac{n}{2}} \left|t_n\left(\frac{1}{2} + ib\right)\right| (θ=bln⁡2\theta = b \ln 2) and a double-angle Blaschke identity −zB(z)=ei2α(b)-z B(z) = e^{i 2\alpha(b)} linking the central polar angle α(b)\alpha(b) of t4(12+ib)t_4(\frac{1}{2} + ib) to its fundamental perimeter L(t4)=Γ(1/4)22πL(t_4) = \frac{\Gamma(1/4)^2}{2\sqrt{\pi}}, thereby parameterizing every individual term tn(12+ib)t_n(\frac{1}{2} + ib) and every concentric dyadic circle on the critical line in terms of the regularized inverse incomplete Beta function Iq(b)−1(14,12)I_{q(b)}^{-1}(\frac{1}{4}, \frac{1}{2}). As a geometric consequence, critical-line zeros ζ(12+ib)=0\zeta(\frac{1}{2} + ib) = 0 are reframed as values of bb where the resultant of the inner concentric circles lands on the antipode −P1(12+ib)-P_1(\frac{1}{2} + ib) of the initial circle of constant radius.

Provenance statement

Google Gemini Argon was used to convert my original figures to more readable versions, produce the LaTeX File, and convert language into a more formal and appropriate form for a mathematical audience.

Tools used

Google
GeminiVersion Argon

References

  1. L. Euler, ``De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt,'' Commentarii Academiae Scientiarum Imperialis Petropolitanae , vol. 5, pp. 36--57, 1730 (published 1738); reprinted in Opera Omnia , Series 1, vol. 14, pp. 1--24.
  2. L. Euler, ``Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques,'' Mémoires de l'Académie des Sciences de Berlin , vol. 17, pp. 83--106, 1768 (read 1749); reprinted in Opera Omnia , Series 1, vol. 15, pp. 70--90.
  3. L. Euler, ``Observationes de comparatione arcuum curvarum irrectificabilium,'' Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae , vol. 6, pp. 58--84, 1761 (read 1751); reprinted in Opera Omnia , Series 1, vol. 20, pp. 80--107.
  4. L. Euler, ``Evolutio formulae integralis integratione a valore x=0x=0 ad x=1x=1 extensa,'' Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae , vol. 16, pp. 91--139, 1772 (read 1771); reprinted in Opera Omnia , Series 1, vol. 17, pp. 316--357.
  5. R. Ayoub, ``Euler and the Zeta Function,'' The American Mathematical Monthly , vol. 81, no. 10, pp. 1067--1086, 1974.
  6. R. Ayoub, ``The Lemniscate and Fagnano's Contributions to Elliptic Integrals,'' Archive for History of Exact Sciences , vol. 29, no. 2, pp. 131--149, 1984.
  7. J. Stillwell, Mathematics and Its History , 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2010.
  8. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis , 4th ed., Cambridge University Press, Cambridge, 1927.
  9. E. C. Titchmarsh and D. R. Heath-Brown, The Theory of the Riemann Zeta-Function , 2nd ed., Oxford University Press, 1986.
  10. H. M. Edwards, Riemann's Zeta Function , Academic Press, New York, 1974.
Show all 14 referencesShow fewer references
  1. T. Needham, Visual Complex Analysis , Oxford University Press, 1997.
  2. J. B. Garnett, Bounded Analytic Functions , Revised 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, 2007.
  3. 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.
  4. T. M. Apostol, Introduction to Analytic Number Theory , Undergraduate Texts in Mathematics, Springer-Verlag, 1976.

Version history

  1. v1 · 16 pagesSubmitted by Ryan HackbarthInitial depositCurrentNov 06, 2026