Rather than analyzing the Riemann zeta function solely as a complex-analytic object, we partition \(\zeta(n)\) into four symmetric sub-series \(h_r(n)\) modulo 4:
\[ h_3(n) = \sum_{k=1}^{\infty} \frac{1}{(4k)^n} = 4^{-n} \zeta(n) \] \[ h_1(n) = \sum_{k=1}^{\infty} \frac{1}{(4k-2)^n} = (2^{-n} - 4^{-n}) \zeta(n) \] \[ h_0(n) = \sum_{k=1}^{\infty} \frac{1}{(4k-3)^n} = 4^{-n} \zeta(n, 1/4) \] \[ h_2(n) = \sum_{k=1}^{\infty} \frac{1}{(4k-1)^n} = 4^{-n} \zeta(n, 3/4) \]To analyze the scaling relationship, we establish the coordinate ratio mapping between the standard view and the alternative view:
| Standard View | Scale | Alternative View | ||
|---|---|---|---|---|
| \(\pi^n\) | \(\cdot\) | \((4/\pi)^n\) | \(=\) | \(4^n\) |
| \(\div\) | \(\div\) | |||
| \(R\) | \(\cdot\) | \((4/\pi)^n\) | \(=\) | \(1/h_3\) |
| \(=\) | \(=\) | |||
| \(\zeta(n)\) | \(\zeta(n)\) |
The alternative coordinate decomposition provides a pristine structural template. Below is the numerical grid at the ground state \(n=4\):
| Position | \(M_0\) | \(M_1\) | \(M_2\) | \(M_3\) | Sum (\(4^n\)) |
|---|---|---|---|---|---|
| Major | \(236.95665276\) | \(15\) | \(3.04334724\) | \(1\) | \(256\) |
| Scale | \(1/h_3\) | \(1/h_3\) | \(1/h_3\) | \(1/h_3\) | \(1/h_3\) |
| Minor | \(h_0\) | \(h_1\) | \(h_2\) | \(h_3\) | \(\zeta(n)\) |
| Numerical Minor | \(1.00181129\) | \(0.06341737\) | \(0.01286674\) | \(0.00422782\) | \(1.082323234\) |
Using the true transcendental values, the geometric partitions are defined as:
\[ \Phi_{\text{pink}}(n) = 2(M_2 - M_3) = 2 \cdot (3.04334724 - 1) = 4.086694473 \] \[ 2(h_2 - h_3) = \zeta(n) \cdot \frac{\Phi_{\text{pink}}(n)}{4^n} = 4.086694473 \cdot h_3 \] \[ \Psi_{\text{purple}}(n) = (M_0 - M_2) - (M_1 + M_3)(M_1 - M_3) = 233.9133055 - 224 = 9.913305527 \] \[ \beta(n) - \eta(n) = \zeta(n) \cdot \frac{\Psi_{\text{purple}}(n)}{4^n} = 9.913305527 \cdot h_3 \]Using the discrete approximation values (where \(M_2 = 3.5\)), the grid collapses to integer-based relations:
\[ \Phi_{\text{salmon}}(n) = 2(M_2 - M_3) = 2 \cdot (3.5 - 1) = 5 \] \[ 2(h_2 - h_3) = \zeta(n) \cdot \frac{\Phi_{\text{salmon}}(n)}{4^n} = 5 \cdot h_3 \] \[ \Psi_{\text{lavender}}(n) = (M_0 - M_2) - (M_1 + M_3)(M_1 - M_3) = 233 - 224 = 9 \] \[ \beta(n) - \eta(n) = \zeta(n) \cdot \frac{\Psi_{\text{lavender}}(n)}{4^n} = 9 \cdot h_3 \]The quotient residue \(K(n)\) represents the ratio of even-parity differences to odd-parity differences:
\[ K(n) = \frac{M_0 - M_2}{M_1 - M_3} = \frac{h_0 - h_2}{h_1 - h_3} = 2^n \cdot \frac{\beta(n)}{\eta(n)} \]For \(n=4\), this ratio is approximately \(16.70809325\). We isolate the transcendental coordinate remainder \(k(n)\) by subtracting the base-2 scale factor:
\[ k(n) = K(n) - 2^n = 2^n \left( \frac{\beta(n)}{\eta(n)} - 1 \right) \approx 0.70809325 \]As \(n \to \infty\), the Dirichlet beta function \(\beta(n) \to 1\). The inverse Riemann zeta function can be asymptotically expanded as \(\frac{1}{\zeta(n)} \approx 1 - 2^{-n}\).
Substituting these limits into our expression simplifies the numerator of the quotient residue to \(4^n(1 - 2^{-n}) = 4^n - 2^n\). Setting \(x = 2^n\), the ratio can be written as a polynomial fraction:
\[ \frac{x^2 - x}{x - 2} \]Applying polynomial division yields:
\[ x + 1 + \frac{2}{x - 2} \]Substituting \(x = 2^n\) back provides the asymptotic form of \(K(n)\):
\[ K(n) \approx 2^n + 1 + \frac{2}{2^n - 2} \]Since \(k(n) = K(n) - 2^n\), we get:
\[ k(n) \approx 1 + \frac{2}{2^n - 2} \]Taking the limit as \(n \to \infty\), the fractional term vanishes, proving convergence:
\[ \lim_{n \to \infty} k(n) = 1 \]We verify this relationship across multiple integer states, showing how \(k(n)\) completely governs the coordinate residues of \(M_0\) and \(M_2\) relative to their integer boundaries:
| \(n\) | \(K(n)\) | \(k(n)\) | \(\text{Term}_{\text{true}}(n)\) | \(\text{Term}_{\text{pred}}(n)\) | \(\text{Difference}\) | \(0.5 + \text{diff}\) | \(0.5 - \text{diff}\) |
|---|---|---|---|---|---|---|---|
| 2 | 4.45472247 | 0.45472247 | 0.45472247 | 0.5 | -0.04527753 | 0.45472247 | 0.54527753 |
| 3 | 8.59811672 | 0.59811672 | 1.79435017 | 1.5 | 0.29435017 | 0.79435017 | 0.20564983 |
| 4 | 16.70809325 | 0.70809325 | 4.95665276 | 4.5 | 0.45665276 | 0.95665276 | 0.04334724 |
| 5 | 32.79127888 | 0.79127888 | 11.86918322 | 11.5 | 0.36918322 | 0.86918322 | 0.13081678 |
| 6 | 64.85290797 | 0.85290797 | 26.44014702 | 26.5 | -0.05985298 | 0.44014702 | 0.55985298 |
| 7 | 128.89761597 | 0.89761597 | 56.54980582 | 56.5 | 0.04980582 | 0.54980582 | 0.45019418 |
| 8 | 256.92945027 | 0.92945027 | 118.04018383 | 118.5 | -0.45981617 | 0.04018383 | 0.95981617 |
We prove that the tail of the coordinate function can never cross an integer threshold, mathematically securing the relation \(\lfloor M_2(n) \rfloor = \lfloor (4/3)^n \rfloor\) for all \(n \ge 2\).
Theorem 1: Let \(M_2(n) = h_2(n)/h_3(n)\) where \(h_2(n) = \sum_{k=1}^\infty (4k-1)^{-n}\) and \(h_3(n) = \sum_{k=1}^\infty (4k)^{-n}\). Then for all \(n \ge 1\):
\[ M_2(n) < \left(\frac{4}{3}\right)^n \]Furthermore, the difference decays exponentially as \(O((2/3)^n)\), guaranteeing \(\lfloor M_2(n) \rfloor = \lfloor (4/3)^n \rfloor\) for all \(n \ge 2\).
Proof: By definition, our sub-series components are:
\[ h_3(n) = 4^{-n} \zeta(n), \quad h_2(n) = 3^{-n} + 4^{-n} \sum_{k=1}^\infty \left(k + \frac{3}{4}\right)^{-n} \]Substituting these into the definition of \(M_2(n)\) yields:
\[ M_2(n) = \frac{1}{\zeta(n)} \left[ \left(\frac{4}{3}\right)^n + \sum_{k=1}^\infty \left( k + \frac{3}{4} \right)^{-n} \right] \]We compute the exact coordinate difference:
\[ \left(\frac{4}{3}\right)^n - M_2(n) = \frac{1}{\zeta(n)} \left[ \left(\frac{4}{3}\right)^n (\zeta(n) - 1) - \sum_{k=1}^\infty \left( k + \frac{3}{4} \right)^{-n} \right] \]Using the identity \(\zeta(n) - 1 = \sum_{k=1}^\infty (k+1)^{-n}\), we scale and substitute:
\[ \left(\frac{4}{3}\right)^n - M_2(n) = \frac{1}{\zeta(n)} \sum_{k=1}^\infty \left[ \left( \frac{4}{3k+3} \right)^n - \left( \frac{4}{4k+3} \right)^n \right] \]Since \(4k+3 > 3k+3 > 0\) for all \(k \ge 1\), each term is strictly positive, proving the strict inequality:
\[ M_2(n) < \left(\frac{4}{3}\right)^n \]The dominant term of the summation decays as \(O((2/3)^n)\), ensuring that the floor is never crossed. This completes the proof. \(\blacksquare\)
This appendix provides the formal translations between the geometric-algebraic coordinate components and standard analytic number theory: