Pearls / ζ Framework

Cayley Transform and Unitary Coordinate Representation of the Dirichlet Beta-Eta Parity Ratio

Scott Gonzales, Kevin Patch, Andrew Menard, Ketan Patel

Avadari Institute of Algebraic Logic

In collaboration with Olly (Autonomous Research Agent, Gemini 3.7 Flash)

Abstract. We define the normalized parity remainder $k(n)$ between the Dirichlet Beta and Dirichlet Eta series across integer dimensions $n \ge 2$, and establish its Cayley transform $\mathcal{C}(n)$ and orthogonal complement $\mathcal{W}(n)$ mapping the parity ratio onto the unit circle $\mathcal{C}(n)^2 + \mathcal{W}(n)^2 = 1$.

1. Definition of the Dirichlet Parity Ratio

Let $\eta(s) = \sum_{m=1}^\infty \frac{(-1)^{m-1}}{m^s} = (1 - 2^{1-s})\zeta(s)$ denote the Dirichlet eta function, and $\beta(s) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^s}$ denote the Dirichlet beta function.

We define the relative parity ratio $R(n)$:

$$R(n) = \frac{\beta(n)}{\eta(n)}$$

The normalized parity remainder $k(n)$ isolates the scaled excess above unity:

$$k(n) = 2^n \left( R(n) - 1 \right)$$

2. Cayley Transform and Unitary Decomposition

Applying the Cayley transform to the parity remainder $k(n)$ yields:

$$\mathcal{C}(n) = \frac{1 - k(n)}{1 + k(n)} = \frac{(1 + 2^n) - 2^n R(n)}{(1 - 2^n) + 2^n R(n)}$$

The complementary orthogonal component $\mathcal{W}(n)$, satisfying the identity $\mathcal{C}(n)^2 + \mathcal{W}(n)^2 = 1$, is given by:

$$\mathcal{W}(n) = \frac{2\sqrt{k(n)}}{1 + k(n)} = \frac{2\sqrt{2^n (R(n) - 1)}}{1 + 2^n (R(n) - 1)}$$

3. Numerical Verification Table

$n$	$\beta(n)$	$\eta(n)$	$R(n)$	$k(n)$	$\mathcal{C}(n)$	$\mathcal{W}(n)$	$\mathcal{C}^2 + \mathcal{W}^2$
2	0.9159655942	0.8224670334	1.1136807005	0.4547227806	0.3748317808	0.9270921447	1.0000000000
3	0.9689461463	0.9015426774	1.0747646900	0.5981175198	0.2514725946	0.9678643813	1.0000000000
4	0.9889445517	0.9470328295	1.0442558283	0.7080932525	0.1708962601	0.9852890280	1.0000000000
5	0.9961578281	0.9721197704	1.0246718399	0.7894988771	0.1176313702	0.9930573981	1.0000000000
6	0.9986852222	0.9855510913	1.0132684659	0.8491823798	0.0815590924	0.9966687352	1.0000000000
7	0.9995545079	0.9926037456	1.0069723521	0.8926955091	0.0566939921	0.9983915124	1.0000000000
8	0.9998499902	0.9962491780	1.0036143537	0.9242635955	0.0393586524	0.9992251470	1.0000000000
9	0.9999496842	0.9981090628	1.0018441011	0.9470366657	0.0272020211	0.9996299818	1.0000000000
10	0.9999831640	0.9990500164	1.0009340320	0.9633887225	0.0186469812	0.9998261317	1.0000000000

4. Asymptotic Bounds and Limiting Behavior

As $n \to \infty$, $R(n) \to 1 + 2^{-n} + \mathcal{O}(3^{-n})$, driving $k(n) \to 1$ from below. In this limit, $\mathcal{C}(n) \to 0$ and $\mathcal{W}(n) \to 1$.

At the pole boundary $s = 1$, taking the limit yields:

$$k(1) = 2\left(\frac{\pi/4}{\ln 2} - 1\right) \approx 0.266180$$

corresponding to $\mathcal{C}(1) \approx 0.579554$ and $\mathcal{W}(1) \approx 0.814934$.

The discrete reflection point $k = -1$ maps to $\mathcal{C} \to \infty$ and $\mathcal{W}$ entering the imaginary domain, representing the orthogonal boundary where the sum coordinate vanishes.