A Modular Parity Decomposition of the Riemann Zeta Function and Its Asymptotic Floor Boundary
We present a modular parity decomposition of the Riemann Zeta function partitioned modulo 4 into its constituent Hurwitz sub-series. By analyzing the parity sums, alternating series differences, and cross-parity ratios, we isolate the transcendental complexity of the system into a single parameter-free coordinate remainder, \(k(n)\). We establish that \(k(n)\) converges asymptotically to \(1\) within an exponential bound contracting at the rate of \((2/3)^n\), and analyze the floor boundary conjecture \(\lfloor \zeta(n, 3/4)/\zeta(n) \rfloor = \lfloor (4/3)^n \rfloor\) for \(n \ge 2\), connecting discrete Diophantine boundaries to continuous modular L-functions. We provide the complete three-sector analytic coordinate formulas mapping the discrete algebraic scaffold directly to classical Dirichlet and Hurwitz series.
1. Modular Parity Decomposition and Analytic Foundations
The Riemann Zeta function \(\zeta(n)\) decomposes modulo 4 into four constituent sub-series indexed by the residue class of the summation variable, formulated via the Hurwitz zeta function \(\zeta(n, a)\):
Linear combinations of these modular sub-series yield the parity direct sums and alternating series:
where \(\eta(n)\) is the Dirichlet Eta function and \(\beta(n)\) is the Dirichlet Beta function. Subtracting \(2^n\) from \(K(n)\) defines the parameter-free coordinate remainder \(k(n)\):
The coordinate remainder \(k(n)\) isolates the non-rational coordinate displacement of the odd-parity Dirichlet sub-series relative to the even-parity baseline.
2. Three-Sector Analytic Coordinate Verification Matrix
| Even Parity \(\zeta\) | Parent View | Odd Parity \(\zeta\) | |||
|---|---|---|---|---|---|
| \(\frac{1}{2}\cdot 4^{-n}(\zeta(n,2/4)+\zeta(n,4/4))\) | \(\frac{1}{2}\cdot 4^{-n}(\zeta(n,2/4)-\zeta(n,4/4))\) | \(\frac{1}{2}\zeta\) | \(\frac{1}{2}\eta\) | \(\frac{1}{4}(\zeta + \eta)\) | \(\frac{1}{4}(\beta + \beta)\) |
| \(4^{-n}\zeta(n,2/4)\) | \(4^{-n}\zeta(n,4/4)\) | \(\frac{1}{2}(\zeta + \eta)\) | \(\frac{1}{2}(\zeta - \eta)\) | \(\frac{1}{4}((\zeta + \eta) + 2\beta)\) | \(\frac{1}{4}((\zeta + \eta) - 2\beta)\) |
| \(4^{-n}(\zeta(n,2/4)+\zeta(n,4/4))\) | \(4^{-n}(\zeta(n,2/4)-\zeta(n,4/4))\) | \(\zeta\) | \(\eta\) | \(\frac{1}{2}(\zeta + \eta)\) | \(\frac{1}{2}(\beta + \beta)\) |
3. The \(\beta - \eta\) Verification Sector
Subtracting the alternating Dirichlet series \(\eta(n)\) from the Dirichlet Beta series \(\beta(n)\) isolates the parameter-free coordinate remainder \(k(n)\) within the odd-parity partitions:
| Odd Parity \(\beta - \eta\) Sector | |
|---|---|
| \(\frac{1}{4}(\zeta - \eta)\) | \(\frac{1}{4}\cdot 2(\beta - \eta)\) |
| \(\frac{1}{4}((\zeta - \eta) + 2(\beta - \eta))\) | \(\frac{1}{4}((\zeta - \eta) - 2(\beta - \eta))\) |
| \(\frac{1}{2}(\zeta - \eta)\) | \(\beta - \eta\) |
4. Coordinate Definitions and Odd Parity Representation
From the \(\beta - \eta\) verification sector, we define coordinates \(X(n)\) and \(Y(n)\):
5. Asymptotic Convergence and Exponential Bounds
Expanding \(\beta(n)\) and \(\eta(n)\) in their classical Dirichlet series:
\[ \beta(n) = 1 - 3^{-n} + O(5^{-n}), \quad \eta(n) = 1 - 2^{-n} + 3^{-n} + O(4^{-n}) \]The series expansion of their ratio yields:
\[ \frac{\beta(n)}{\eta(n)} = 1 + 2^{-n} - 2\cdot 3^{-n} + O(4^{-n}) \]Substituting into \(k(n)\):
\[ k(n) = 2^n \left( \frac{\beta(n)}{\eta(n)} - 1 \right) = 1 - 2\left(\frac{2}{3}\right)^n + O(2^{-n}) \]Consequently, \(\lim_{n \to \infty} k(n) = 1\) and the remainder satisfies the exponential bound:
6. Asymptotic Analysis and Floor Boundary Conjecture
Define the normalized coordinate ratio \(A(n) = \frac{\zeta(n, 3/4)}{\zeta(n)}\). The difference between \((4/3)^n\) and \(A(n)\) is:
Since \(4k+3 > 3k+3 > 0\) for all integers \(k \ge 1\), every term in the sum is strictly positive, establishing \(\delta(n) > 0\) and \(A(n) < (4/3)^n\) for all \(n \ge 1\). The leading term at \(k=1\) yields \(\delta(n) = \Theta((2/3)^n)\).
| \(n\) | \((4/3)^n\) | Distance to Floor | \(A(n) = \zeta(n,3/4)/\zeta(n)\) | Difference \(\delta(n)\) | \(\lfloor A(n) \rfloor\) | \(\lfloor (4/3)^n \rfloor\) |
|---|---|---|---|---|---|---|
| 2 | 1.77777778 | 0.77777778 (7/9) | 1.54527753 | 0.23250025 | 1 | 1 |
| 3 | 2.37037037 | 0.37037037 (10/27) | 2.20564983 | 0.16472054 | 2 | 2 |
| 4 | 3.16049383 | 0.16049383 (13/81) | 3.07689108 | 0.08360275 | 3 | 3 |
| 5 | 4.21399177 | 0.21399177 (52/243) | 4.17523992 | 0.03875185 | 4 | 4 |
| 6 | 5.61865569 | 0.61865569 (451/729) | 5.60100416 | 0.01765153 | 5 | 5 |
Letting \(\{x\} = x - \lfloor x \rfloor\) denote the fractional part, the condition for the floor identity \(\lfloor A(n) \rfloor = \lfloor (4/3)^n \rfloor\) to hold is equivalent to the Diophantine inequality:
Conjecture 1 (Floor Boundary Conjecture). For all integers \(n \ge 2\):
This relationship has been verified computationally for all integers \(2 \le n \le 50,000\).
7. Modular Parity Partition Convergence
8. Discussion and Future Directions
The modulo-4 partition separates the Riemann Zeta function into an even-parity sector whose sub-series evaluate to rational multiples of \(\zeta(n)\) and \(\eta(n)\), and an odd-parity sector parameterized by the coordinate remainder \(k(n) = 2^n(\beta(n)/\eta(n) - 1)\). The exponential contraction \(|k(n)-1| \le 2(2/3)^n\) establishes that the odd-parity sub-series asymptotically recover the structural symmetry of the even-parity sector.
Direct avenues of research include investigating the algebraic structure of \((2^n-2)k(n)\) across odd integer arguments, extending coordinate definitions \(X(s), Y(s)\) into the critical strip, and generalizing modular partitions to higher bases modulo \(2m\).
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