math.CV — Complex Variables

Origin-Centered Mobius Circles, Bernoulli Lemniscates, and Beta–Gamma Dyadic Pairings of the Riemann Zeta Function on the Critical Line

Contributed by Ryan Hackbarth

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