Prime Number Theorem Proof via Mean Value Theorem: A Simplified Approach — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The search results are dominated by expository videos on prime distribution and the Riemann Hypothesis, with only one substantive paper (arXiv:1510.03465) offering a simplified proof of the Prime Number Theorem (PNT) via the Mean Value Theorem for arithmetic functions. No new "hidden pattern" or "Prime Scalar Field" mathematical content is provided beyond standard theory. | MATH: The core mathematics remains the Riemann zeta function ζ(s) = Σ n⁻ˢ = Π (1 − p⁻ˢ)⁻¹ (Euler product), the Prime Number Theorem π(x) ~ Li(x) ~ x/ln(x), and the Riemann Hypothesis: all non-trivial zeros of ζ(s) lie on the critical line Re(s) = 1/2. The arXiv paper's contribution is purely methodological — using the Mean Value Theorem to derive π(x) ~ x/ln(x) without deeper analytic machinery. No new constants, ratios, or equations emerge. | CONNECTION: The critical line Re(s) = 1/2 is the only direct geometric link — a symmetry axis in the complex plane. The trivial zeros at negative even integers (−2, −
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026prime,
title = {{Prime Number Theorem Proof via Mean Value Theorem: A Simplified Approach — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23526049},
url = {https://zenodo.org/records/23526049},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Prime Number Theorem Proof via Mean Value Theorem: A Simplified Approach — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23526049
UR - https://zenodo.org/records/23526049
AB - FINDING: The search results are dominated by expository videos on prime distribution and the Riemann Hypothesis, with only one substantive paper (arXiv:1510.03465) offering a simplified proof of the Prime Number Theorem (PNT) via the Mean Value Theorem for arithmetic functions. No new "hidden pattern" or "Prime Scalar Field" mathematical content is provided beyond standard theory. | MATH: The core mathematics remains the Riemann zeta function ζ(s) = Σ n⁻ˢ = Π (1 − p⁻ˢ)⁻¹ (Euler product), the Prime Number Theorem π(x) ~ Li(x) ~ x/ln(x), and the Riemann Hypothesis: all non-trivial zeros of ζ(s) lie on the critical line Re(s) = 1/2. The arXiv paper's contribution is purely methodological — using the Mean Value Theorem to derive π(x) ~ x/ln(x) without deeper analytic machinery. No new constants, ratios, or equations emerge. | CONNECTION: The critical line Re(s) = 1/2 is the only direct geometric link — a symmetry axis in the complex plane. The trivial zeros at negative even integers (−2, − Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -