Mathematical Pearl · Reference Note / Zeta

Modulo-4 Analytic Partitions: Bridging Euler's Exponential Decompositions to Dirichlet Series and the Coordinate Ratio $K(s)$

Scott Gonzales, Kevin Patch, Andrew Menard, Ketan Patel*

Avadari Institute of Algebraic Logic

* Corresponding author. Formalization, computational verification, and analytical drafting conducted in collaboration with an Autonomous Research Agent (Olly: Gemini 3.7 Flash).

DOI: 10.5281/zenodo.23198232

Abstract. We present a direct structural correspondence between Euler's modulo-4 partition of the exponential series and the modulo-4 arithmetic partition of Dirichlet series. By evaluating the symmetric sums and alternating differences of both systems, we demonstrate that the rational factor $2^s - 1$ emerges as the Dirichlet analogue of the hyperbolic partition ratio, while the Coordinate Ratio $K(s)$ emerges as the Dirichlet analogue of the circular cotangent partition ratio, defined by the quotient of the Dirichlet beta function and the scaled Dirichlet eta function.

1. Modulo-4 Partition of the Exponential Generating Function

Euler's decomposition of the exponential function $e^x$ partitions the power series into four subseries according to the residue modulo 4 of the summation index:

$$S_0(x) = \sum_{k=0}^{\infty} \frac{x^{4k}}{(4k)!}$$ $$S_1(x) = \sum_{k=0}^{\infty} \frac{x^{4k+1}}{(4k+1)!}$$ $$S_2(x) = \sum_{k=0}^{\infty} \frac{x^{4k+2}}{(4k+2)!}$$ $$S_3(x) = \sum_{k=0}^{\infty} \frac{x^{4k+3}}{(4k+3)!}$$

Evaluating the parity sums yields the hyperbolic functions:

Even parity sum: $$S_0(x) + S_2(x) = \cosh(x)$$
Odd parity sum: $$S_1(x) + S_3(x) = \sinh(x)$$

Their ratio defines the hyperbolic cotangent:

$$\frac{S_0(x) + S_2(x)}{S_1(x) + S_3(x)} = \coth(x)$$

Evaluating the alternating differences yields the circular trigonometric functions:

Even alternating difference: $$S_0(x) - S_2(x) = \cos(x)$$
Odd alternating difference: $$S_1(x) - S_3(x) = \sin(x)$$

Their ratio defines the circular cotangent:

$$\frac{S_0(x) - S_2(x)}{S_1(x) - S_3(x)} = \cot(x)$$

2. Modulo-4 Partition of Dirichlet Series

We apply the identical modulo-4 partition to the Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ for $\operatorname{Re}(s) > 1$, expressed via Hurwitz zeta functions $\zeta(s, a)$:

$$Z_1(s) = \sum_{k=0}^{\infty} (4k+1)^{-s} = 4^{-s} \zeta\left(s, \frac{1}{4}\right)$$ $$Z_2(s) = \sum_{k=0}^{\infty} (4k+2)^{-s} = 2^{-s} \sum_{m=0}^{\infty} (2m+1)^{-s} = 2^{-s} (1 - 2^{-s}) \zeta(s)$$ $$Z_3(s) = \sum_{k=0}^{\infty} (4k+3)^{-s} = 4^{-s} \zeta\left(s, \frac{3}{4}\right)$$ $$Z_0(s) = \sum_{k=1}^{\infty} (4k)^{-s} = 4^{-s} \zeta(s)$$

Parity Sums (The Hyperbolic Analogue)

Odd parity sum: $$Z_1(s) + Z_3(s) = \sum_{n \text{ odd}} n^{-s} = (1 - 2^{-s}) \zeta(s)$$
Even parity sum: $$Z_2(s) + Z_0(s) = \sum_{n \text{ even}} n^{-s} = 2^{-s} \zeta(s)$$

The ratio of the odd to even parity sums cancels the transcendental component $\zeta(s)$ identically:

$$\frac{Z_1(s) + Z_3(s)}{Z_2(s) + Z_0(s)} = \frac{1 - 2^{-s}}{2^{-s}} = 2^s - 1$$

Alternating Differences (The Circular Analogue and Coordinate Ratio $K(s)$)

Odd alternating difference: $$Z_1(s) - Z_3(s) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^s} = \beta(s)$$
where $\beta(s)$ is the Dirichlet beta function.
Even alternating difference: $$Z_2(s) - Z_0(s) = 2^{-s} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} = 2^{-s} \eta(s)$$
where $\eta(s) = (1 - 2^{1-s}) \zeta(s)$ is the Dirichlet eta function.

The ratio of the odd to even alternating differences defines the Coordinate Ratio $K(s)$:

$$K(s) = \frac{Z_1(s) - Z_3(s)}{Z_2(s) - Z_0(s)} = \frac{\beta(s)}{2^{-s} \eta(s)} = 2^s \frac{\beta(s)}{\eta(s)}$$

3. The Coordinate Remainder $k(s)$

The Coordinate Remainder $k(s)$ isolates the transcendental deviation of the alternating ratio $K(s)$ relative to the rational base scale $2^s$:

$$k(s) = K(s) - 2^s = 2^s \left( \frac{\beta(s)}{\eta(s)} - 1 \right)$$

4. Conclusion

Through this partition, Euler's exponential framework and the Dirichlet series framework are revealed as dual representations of a single modulo-4 symmetry. The scalar $2^s - 1$ represents the pure geometric parity partition, while $K(s)$ and $k(s)$ govern the phase-shifted alternating spectrum.