The Riemann Hypothesis: A Proof via Inverse Sieve of Eratosthenes

This paper presents a formal mathematical argument claiming a proof of the Riemann Hypothesis (RH) through a framework called Arithmetic Spectral Theory (AST). The approach connects the distribution of prime numbers to the non-trivial zeros of the Riemann zeta function $\zeta(s)$ by treating prime sequences as discrete signals and analyzing them in the frequency domain.

Executive Summary of the Proof

The core of the argument relies on constructing a bridge between the classic Sieve of Eratosthenes and the spectral properties of linear operators. The proof is structured around three main pillars:

1. Signal Construction: The prime numbers are represented as a discrete binary indicator function $1_{\mathbb{P}}(n)$ on the natural numbers.

2. Spectral Extraction: The Fourier transform of this prime indicator function, $F(\gamma)$, is shown to share identical frequency characteristics and peak locations with a weighted prime sum $G(\gamma)$. Because $G(\gamma)$ explicitly contains poles corresponding to the zeros of $\zeta(s)$, the peaks in the Fourier spectrum map directly to the imaginary parts ($\gamma$) of the non-trivial zeros.

3. Operator Self-Adjointness: Because the prime indicator function is entirely real-valued, its Fourier transform exhibits perfect conjugate symmetry ($\overline{F(\gamma)} = F(-\gamma)$). This symmetry implies that the corresponding multiplication operator $\hat{F}$ is self-adjoint on $L^{2}(\mathbb{R}, d\gamma)$. By the spectral theorem, self-adjoint operators possess strictly real eigenvalues. Consequently, all extracted $\gamma$ values must be purely real, forcing all non-trivial zeros to lie precisely on the critical line $Re(s) = 1/2$.

Methodological Breakdown

1. The Fourier Transform & Weighted Prime Sum

The paper defines the standard Fourier transform of the prime positions as:

It contrasts this with a weighted prime sum derived from the logarithmic derivative of the zeta function:

The text argues that the weights ($\frac{\log p}{\sqrt{p}}$) affect only the amplitudes of the signal's components, leaving the underlying oscillation frequencies unchanged. As a result, constructive interference creates spectral peaks at the exact same $\gamma$ values for both functions, linking the eigenvalues of the operator directly to the zeta zeros.

2. The L-EFM Operator

The proof incorporates the Laplace-Euler-Fourier-Mellin (L-EFM) operator, defined as an extension of the Euler product to the critical line:

This operator uses a half-line restriction ($u \ge 0$) to ensure the exponential decay of $e^{-\sigma u}$, establishing "half-line admissibility" within generalized function spaces.

Numerical Verification & Methodology

To validate the theoretical framework, the paper details a deterministic, reproducible algorithmic pipeline:

• Signal Scope: A binary signal is constructed using the first $N = 200,000$ natural numbers, capturing exactly 17,984 primes.

• Scale Calibration: Because the fast Fourier transform (FFT) outputs discrete frequencies ($f$), a scaling factor ($\alpha$) is required to map them to the imaginary parts of the zeros ($\gamma = \alpha f$).

  • A Global Scale Calibration optimizes a single factor ($\alpha_{global} = 116.28$) to minimize the total absolute error across the first 10 known zeros.

  • An Individual Scale Refinement then executes fine sweeps to isolate individual peaks.

• Results: Using an arbitrary reproducibility seed of 123, the individual refinement methodology achieves a 100% match rate ($15/15$) for the first fifteen non-trivial zeros of the zeta function, yielding a mean error of $0.000000055$ and a maximum error of $0.000000125$.

Broader Framework Applications

The paper notes that this spectral approach to prime numbers is not isolated to the Riemann Hypothesis, detailing two major parallel applications:

Pure Mathematics: Quantified Green-Tao Theorem

Using a prime kernel consisting of the first six primes $\{2, 3, 5, 7, 11, 13\}$, the L-EFM operator calculates the spectral coherence of prime arithmetic progressions. The analysis shows a monotonic decrease in coherence as progression lengths increase (from 0.8746 for length-3 progressions down to 0.7398 for length-6). This establishes a spectral law indicating that longer prime progressions possess inherently lower spectral coherence.

Applied Science: AI Safety (H2E Sheriff)

The prime-anchored coherence calculations are adapted to agentic AI systems within the Human-to-Expert (H2E) framework. By utilizing a five-prime kernel, the system derives a universal mathematical safety threshold ($\Lambda = 0.9583$). This boundary acts as a deterministic "hard-stop" layer to govern AI behavior and prevent catastrophic forgetting in prime-anchored neural networks.