Divisor Wave Plotter

📄 Divisor Wave Product Analysis of Prime and Composite Numbers

📄 Divisor Wave Product Analysis of Prime and Composite Numbers (new version — in active development)

An Electron + Next.js desktop app for generating 3D surface plots of infinite-product functions over the complex plane. The functions are motivated by a connection between divisor waves, prime number distribution, and the Riemann Hypothesis.

Product of Product Representation for Sin — Normalized, Jet colormap, x∈[2,20], y∈[-5,5]

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Mathematical Background

A divisor wave of order k is a real-variable function of the form:

$$a_k(x) = \left| \alpha \frac{x}{k} \sin!\left(\frac{\pi x}{k}\right) \right|$$

where k is a positive integer greater than 1. Its zeros fall at every integer multiple of k, and its amplitude grows as α(x/k). Taking the infinite product over all k from 2 to x gives a(z):

$$a(z) = \left|\prod_{k=2}^{x} \alpha\frac{x}{k}\sin!\left(\frac{\pi z}{k}\right)\right|$$

a(z) is zero at both primes and composites, but the shape of each zero differs: at a prime p only one wave a_p has an x-intercept there, so the function touches zero sharply and immediately returns — a cusp. At a composite, multiple divisor waves share the zero, pulling the function down more gradually — a curve.

Substituting the Weierstrass product representation of sin into a(z) yields b(z):

$$b(z) = \left|\prod_{k=2}^{x} \frac{\beta x}{k}\left(\pi z \prod_{n=2}^{x}\left(1 - \frac{z^2}{n^2 k^2}\right)\right)\right|$$

b(z) has better left- and right-hand limits at each whole number than a(z) does, and this is what produces the key property: the factor $(1 - z^2/n^2k^2)$ is zero only when $z = nk$, i.e. when z is a product of two integers each ≥ 2 — a composite. For a prime p, no pair (n, k) with n, k ≥ 2 satisfies nk = p, so none of the factors vanish and b(p) ≠ 0. For any composite c = nk, the corresponding factor is exactly zero and b(c) = 0. b(z) is therefore a continuous prime indicator — nonzero at primes, zero at composites.

Extending both functions to the complex plane and composing with other infinite-product representations (Riesz, Viète) is the central object of study.

The functions are rendered as 3D surfaces over the complex plane: the real axis runs left-right, the imaginary axis runs front-back, and the z-axis shows the function's magnitude.

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Structure

python-divisor-wave/        # matplotlib 3D renderer and function library
  special_functions.py      # all 32 infinite-product functions
  plot_cli.py               # headless CLI, spawned by Electron
  divisor_wave_plotting.py  # original PyQt5 GUI (legacy)

electron-divisor-wave/      # Electron shell + Next.js UI
  pages/index.js            # full React UI
  electron/main.js          # IPC bridge to Python CLI
  electron/preload.js       # context-isolated API surface

c-sharp-plotting/           # C# alternative plotter
latex-divisor-wave-paper/   # LaTeX paper: "Divisor Wave Product Analysis of Prime and Composite Numbers"
plot-outputs/               # saved PNG renders (gitignored)

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Running

Requirements: Python 3, Node.js

# 1. Python dependencies (one-time)
python -m venv venv
venv\Scripts\activate        # Windows
# source venv/bin/activate   # macOS / Linux
pip install numpy matplotlib scipy mpmath

# 2. Node dependencies (one-time)
cd electron-divisor-wave
npm install

# 3. Launch
npm run dev

The Electron window opens automatically once Next.js is ready on http://localhost:3000. Plots are saved as PNGs to plot-outputs/.

Headless CLI

The Python backend can also be used directly:

python python-divisor-wave/plot_cli.py \
  --function 2 --normalize Y --colormap 2 \
  --resolution 0.05 --xmin 2 --xmax 20 --ymin -5 --ymax 5 \
  --elev 30 --azim -45 --output ./plot-outputs
# prints: {"success": true, "path": "...", "name": "..."}

# Auto-fit coefficients without rendering:
python python-divisor-wave/plot_cli.py \
  --function 2 --normalize Y --auto-coeffs
# prints: {"success": true, "m": 0.36, "beta": 0.147, ...}

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Functions

32 functions in 7 groups, all evaluated pointwise over a meshgrid of complex numbers.

Basic

#	Name	Formula
1	Product of Sin	$\prod_{n=2}^{x} |\beta\tfrac{x}{n}\sin(\pi z/n)|^{-m}$
2	Product Rep. for Sin	$\prod_{n=2}^{x} |\beta\tfrac{x}{n} \cdot \pi z \prod_{k=2}^{x}(1-z^2/n^2k^2)|^{-m}$
4	Complex Playground Demo	Riesz-cos variant, curated for visual interest

Riesz Products

#	Name	Formula
5	Riesz — Cos	$\prod_{n=2}^{x} |iz_y + \cos(\pi z n)|^{iz_y \cdot (-m)}$
6	Riesz — Sin	$\prod_{n=2}^{x} |1 + \sin(\pi z n)|^{-m}$
7	Riesz — Tan	$\prod_{n=2}^{x} |1 + \tan(\pi z n)|^{-m}$

Viète Products

#	Name	Formula
8	Viète — Cos	$\prod_{n=2}^{x} \left|\cos!\left(\dfrac{\pi z}{2^n}\right)\right|^{-m}$
9	Viète — Sin	$\prod_{n=2}^{x} \left|\sin!\left(\dfrac{\pi z}{2^n}\right)\right|^{-m}$
10	Viète — Tan	$\prod_{n=2}^{x} \left|\tan!\left(\dfrac{\pi z}{2^n}\right)\right|^{-m}$

These converge quickly — large m values are typically needed for Cos and Sin; very small m for Tan, which diverges easily.

Compositions

Outer trig applied to the product-of-sin or product-rep-for-sin:

• 11: cos(Product of Sin)
• 12: sin(Product of Sin)
• 13: cos(Product Rep. for Sin)
• 14: sin(Product Rep. for Sin)

Prime Indicators

• 15 — Binary Prime Indicator H: $H(z) = P_1(z)^{P_2(z)}$ — combines product-of-sin and product-rep-for-sin to produce peaks at primes.
• 16 — Prime Output Indicator J: $J(z) = P_1^{P_1^{P_2}}$ — iterated tower form.
• 17 — BOPIF Q Alternation Series: Alternating product $(-1)^{H(z)}$ composed with H and J.
• 18 — Dirichlet Eta from BOPIF: $\prod(1 + Q \cdot J)^{-c}$ — motivated by the Dirichlet eta function.

Analytic Baselines

Standard complex analysis references for comparison:

• 19: $|\log\Gamma(z)|$
• 20: $1/(1+z^2)$
• 21: $|z^z|$
• 22: $\Gamma(z)$

Transforms / Custom / Experimental

• 23–25: Log, Gamma, and Gamma-form transforms of the product rep.
• 26–27: Custom Riesz–Tan and Viète–Cos with gamma normalization.
• 28: Half-Base Viète–Sin (denominator $2^{-n}$ instead of $2^n$).
• 29: Log-Power-Base Viète–Sin (denominator $2^{\frac{\log z}{n}}$).
• 30: Riesz–Tan combined with the prime indicator.
• 31: Nested roots / paradox product for 2.

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Coefficients

Each function is scaled by two parameters:

Param	Role
m	Exponent applied to the whole product — controls vertical stretch. Smaller = flatter surface; larger = taller peaks.
β	Lead amplitude coefficient inside each product term.

Each function ships with hand-tuned defaults. The auto button samples the function on a coarse grid, computes log-mean and log-variance, and fits m and β using a Lyapunov-derived heuristic:

m_opt   = 1 / std(log|output|)
beta_opt = exp(mean(log|output|) / N_avg)

The override button lets you dial them in manually with sliders.

Normalization divides the product by $\Gamma(\text{result})$ before rendering, compressing large spikes and revealing finer structure.

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Controls

Control	Description
Function	Select from 32 functions grouped by type
Normalization	Raw or Normalized (÷Γ)
Colormap	Prism · Jet · Plasma · Viridis · Magma
Resolution	Grid step — smaller = finer & slower. ~80k points triggers a warning.
Domain	X min/max (real axis), Y min/max (imaginary axis)
View Angles	Elevation and azimuth for the 3D camera
Coefficients	Per-function m and β, with auto-fit and manual override
History	Last 20 renders shown as a strip at the bottom; click to reload

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Paper

The accompanying LaTeX paper — "Divisor Wave Product Analysis of Prime and Composite Numbers" by Leo J. Borcherding — develops the theory behind these functions and their relationship to the distribution of primes. See latex-divisor-wave-paper/.