Fejér--Kernel Prime Indicators

Author: Sebastian Fuchs
Email: sebastian.fuchs@hu-berlin.de
Location: Berlin, Germany
Affiliation: Humboldt University of Berlin
GitHub: SebastianFoxxx/analytic-prime-indicator
Paper on arXiv: https://arxiv.org/abs/2506.18933
ORCID: 0009-0009-1237-4804
DOI: 10.5281/zenodo.15748475
Date: 2025-10-15
Version: 1.2.0

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Abstract

A $C^1$ prime indicator $\mathcal{P}: \mathbb{R} \to \mathbb{R}$ is constructed by applying the Fejér identity to the sine–quotient encoder of trial division. For integers $n\ge 2$, $\mathcal P(n)=0$ holds exactly for odd primes; $\mathcal P(2)>0$. For all non-integers $x>1$ one has $\mathcal P(x)>0$. The function is piecewise $C^\infty$ and its second derivative has jumps precisely at the squares $m^2$, with explicit sizes. Replacing the sharp cut-off by a smooth transition yields $C^\infty$ analogues $\mathcal{P}\tau$ and $\mathcal{P}\sigma$ with integer limits $\mathcal{P}\tau(n;\kappa)\to \tau(n)-2$ and $\mathcal{P}\sigma(n;\kappa)\to \sigma(n)-n-1$ as $\kappa\to\infty$, obtained from locally uniform convergence of derivative series. For large $\kappa$, numerical evidence indicates companion zeros near odd primes for $\mathcal{P}\tau$ and an asymmetric pair for $\mathcal{P}\sigma$. No assertion is made beyond integer input, and no statements are claimed about the prime number theorem or zero distributions of $L$-functions. The appendix includes two illustrative prime-counting sums.

Numerical Verification and Plotting Script

The script numerical_verification.py is a comprehensive suite for the numerical work presented in the paper. It is designed to be highly configurable, performant, and robust. Its key features include:

• Core Implementations: Provides performant implementations of the foundational function $\mathcal{P}(x)$ and its smooth analogues $\mathcal{P}{\tau}(x)$ and $\mathcal{P}{\sigma}(x)$, accelerated with Numba for just-in-time compilation.
• Numerical Verification Suite: A series of rigorous tests to validate the properties of the functions:
  • High-Precision Zero Checks: Verifies that $\mathcal{P}(n)=0$ exclusively for odd primes at integer arguments using the arbitrary-precision mpmath library.
  • Positivity Scans: Numerically confirms that $\mathcal{P}(x) > 0$ for non-integer $x > 1$ on a fine-grained grid.
  • Consistency Tests: Ensures that different evaluation methods for the core Fejér term (closed-form, cosine polynomial, and resonant partial fractions) are numerically consistent within specified tolerances.
  • Companion Zero Validation: Tests the conjectures regarding the behavior of zeros for $\mathcal{P}{\tau}(x)$ and $\mathcal{P}{\sigma}(x)$ near odd primes for varying steepness parameters.
• Performance Optimization: Leverages multiprocessing via Python's multiprocessing pool to significantly speed up computationally intensive verification tasks.
• Plot Generation: Automatically generates all figures presented in the paper in publication-quality format (.pdf by default). This includes overviews, local zooms, derivative plots, and visualizations of the smooth analogues and their properties.
• Central Configuration: All numerical parameters, test ranges, tolerances, and plot settings can be easily adjusted within the central Config class at the top of the script.

Repository Contents

This repository provides all the necessary materials to reproduce the results and figures presented in the paper. The structure is as follows:

• /paper/: Contains the LaTeX source code (analytic_prime_indicator.tex) and all figures for the main manuscript.
• /code/: Contains the primary Python script for verification and plotting.
  • numerical_verification.py: The script to run all numerical tests and generate all figures.
  • requirements.txt: A list of all required Python packages.
• LICENSE: The license under which the code in this repository is shared (MIT License).
• .gitignore: Specifies files to be ignored by version control.

How to Reproduce the Results

To run the numerical verifications and regenerate all plots from the paper, please follow these steps.

Prerequisites

• Python 3.8 or newer.
• A Python package manager, pip.
• A functioning LaTeX distribution (e.g., TeX Live, MiKTeX).

Step-by-Step Instructions

1. Clone the Repository Clone this repository to your local machine using:

   git clone https://github.com/SebastianFoxxx/analytic-prime-indicator.git
   cd analytic-prime-indicator

2. Set Up a Virtual Environment (Recommended) It is best practice to create a virtual environment to avoid conflicts with other Python projects.

   # For Linux/macOS
   python3 -m venv venv
   source venv/bin/activate
   
   # For Windows
   # python -m venv venv
   # venv\Scripts\activate

3. Install Required Packages Install all dependencies listed in requirements.txt using pip.

   pip install -r code/requirements.txt

4. Run the Verification Script Execute the main Python script. This will perform all numerical checks described in the script and regenerate all figures (.pdf files) in the directory where the script is located.

   python code/numerical_verification.py

   The script will print its progress to the console.

5. Compile the LaTeX Document To generate the PDF from the source, compile the .tex file using your LaTeX distribution. You may need to run the command twice for cross-references to be resolved correctly.

   # To compile the main paper
   pdflatex -output-directory=paper paper/analytic_prime_indicator.tex

License

The source code in this repository is released under the MIT License. See the LICENSE file for more details. The content of the research paper is subject to the copyright of the author.