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March 3, 20260 citationsOpen Access

The Sieve Firewall / The True Riemann Hypothesis and Request Solved!

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DSDaniel Santiago

Key Points

  • This research aims to reformulate and computationally verify Riemann's hypothesis about the zeros of the zeta function.
  • Developed a formal framework to reduce Riemann's observation to an anticorrelation bound (S1*)
  • Implemented the NOXA computational engine to track periodic terms across multiple scans
  • Conducted 1,628 scans analyzing 130,240 zeros over a sieve window up to x ≈ 9.5 million
  • S1* was validated in every scan with the ratio decreasing from 0.000862 to 0.000216
  • Lemma 8 and its generalization were proven unconditionally
  • Six independent AI systems confirmed the mathematical framework

Abstract

A structural reformulation and computational verification of Bernhard Riemann's 1859 hypothesis about the zeros of the zeta function. Riemann observed that the roots of ξ(t) appear to be real, wished for a stricter proof, and closed his paper asking future mathematicians to track the influence of the individual periodic terms on the density of primes. This paper answers all three requests. The formal framework (Parts I–VII) reduces Riemann's observation to a single integrated anticorrelation bound (S1*), proven equivalent to RH in both directions. Lemma 8 (Pair Positivity) and its k-body generalization (Corollary 8.1, cost (k−1)² matching GUE Vandermonde repulsion) are proven unconditionally. The NOXA computational engine (Parts VIII–IX, XI) directly implements Riemann's closing request, tracking the periodic terms across 1,628 scans (130,240 zeros, sieve windows to x ≈ 9.5 million). S1* holds in every scan with the ratio declining from 0.000862 to 0.000216. The paper distinguishes Riemann's actual hypothesis (an empirical observation) from the Clay Millennium Prize reformulation (a theorem demand formulated by later mathematicians). Six independent AI systems have cross-validated the mathematical framework.

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Cite This Study

Daniel Santiago (2026) studied this question.

synapsesocial.com/papers/69a67ed1f353c071a6f0a5adhttps://doi.org/10.5281/zenodo.18827898
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