Investigation about a statement equivalent to Riemann Hypothesis (RH)

1. Problem Statement

The paper addresses the Riemann Hypothesis (RH), which posits that all non-trivial zeros of the Riemann zeta function, $\zeta(s)$, lie on the critical line $\Re(s) = 1/2$.
The author focuses on a known equivalence to RH: All local maxima of the function $\xi(t) = \xi(1/2 + it)$ are positive, and all local minima are negative.
The paper aims to:

1. Prove this equivalence using a physical analogy involving "angular momentum."
2. Demonstrate that this positivity condition holds for the $\xi$ function using the Euler Product and the Prime Number Theorem (PNT).
3. Show that if this condition holds, zeros off the critical line must be excluded, thereby proving RH.

2. Methodology

The author employs a multi-faceted approach combining complex analysis, number theory, and a novel physical interpretation:

• Angular Momentum Analogy:

  • The author defines a quantity analogous to angular momentum, $L[\xi(s)]$, for a unitary mass at coordinates $(\Re[\xi(s)], \Im[\xi(s)])$ as time $t = \Im(s)$ varies.
  • $L[\xi(s)]$ is defined as the determinant of the position and velocity vectors, which simplifies to $|\xi(s)|^2 \frac{\partial \angle[\xi(s)]}{\partial t}$.
  • The core hypothesis is that the derivative of this angular momentum with respect to the distance from the critical line ($\epsilon = \Re(s) - 1/2$) at $\epsilon=0$ is strictly positive.

• Phase Variation via Euler Product:

  • Instead of relying solely on the series definition of $\zeta(s)$, the author analyzes the phase variation of $\ln(\zeta(s)(s-1))$ using the Euler Product representation.
  • The paper investigates the convergence of the Euler product in the critical strip ($0 < \Re(s) < 1$), specifically for $\Re(s) > 1/2$.

• Integral Transforms and Prime Distribution:

  • The author utilizes the relationship between $\zeta(s)$ and the prime counting function $J(x)$ (and its approximation $\text{Li}(x)$) via Mellin transforms.
  • A key technique involves calculating the mean phase variation over an interval $\Delta t$ defined by a prime $p^*$, effectively creating a "spectral" analysis of prime numbers.

• Lemma-Based Proofs:

  • The proof is structured around a series of lemmas establishing the convergence of specific integrals involving the Euler product and the difference $J(x) - \text{Li}(x)$.

3. Key Contributions and Technical Results

A. Equivalence to Angular Momentum (Section 2)

The author establishes that the known RH equivalence (positivity of maxima/minima of $\xi(t)$) is mathematically equivalent to the condition:
$$\left[ \frac{\partial L[\xi(t, \epsilon)]}{\partial \epsilon} \right]_{\epsilon=0} = [\xi'(t)]^2 - \xi(t)\xi''(t) > 0$$
This quantity represents the rate of change of the "angular momentum" as one moves away from the critical line. If this derivative is positive everywhere (where $\xi(t) \neq 0$), it implies the specific behavior of local extrema required by RH.

B. Convergence of Euler Product in the Critical Strip (Section 3)

A significant portion of the paper is dedicated to proving that the Euler product representation can be extended into the critical strip ($\Re(s) > 1/2$) under specific conditions:

• Lemma 2: Proves that the expression $\ln[\text{EP}(s)] - \int_0^\infty x^{-s} d[\text{Li}(x)]$ converges absolutely for $\epsilon > 0$ (i.e., $\Re(s) > 1/2$).
• Lemma 2A & 2B: These lemmas demonstrate that the remainder term $R(s, p_{max})$ in the finite Euler product approximation tends to zero as $p_{max} \to \infty$ for $\epsilon > 0$. This validates the use of the Euler product to analyze the phase of $\zeta(s)$ within the critical strip.

C. The "Spectrum of Primes" and Phase Derivative (Section 4)

The author derives an expression for the phase variation $\frac{\partial \angle[\xi(s)]}{\partial t}$ using the Euler product and the difference between the actual prime power distribution $J(x)$ and the logarithmic integral $\text{Li}(x)$.

• Lemma 4: The author estimates the sign of the mixed derivative $\frac{\partial^2 \Im[\ln(\zeta(s)(s-1))]}{\partial \epsilon \partial t}$.
• Theorem 1: By combining the convergence results with the properties of the angular momentum, the author concludes:
  $$\left( \frac{d\xi(t)}{dt} \right)^2 - \xi(t) \frac{d^2\xi(t)}{dt^2} > 0$$
  for all $t$ where $\xi(t) \neq 0$.

D. Numerical and Analytical Support

• Appendices: The paper provides numerical comparisons showing the convergence of the derived integral expressions (Eq 3.21) and evaluates the integral in the critical strip.
• Gibbs Phenomenon: The author addresses potential oscillation issues (Gibbs phenomenon) arising from finite prime limits by using moving windows, ensuring the validity of the convergence intervals.

4. Results

1. Proof of Positivity: The paper claims to prove that the quantity $[\xi'(t)]^2 - \xi(t)\xi''(t)$ is strictly positive for all $t$ where $\xi(t) \neq 0$.
2. Exclusion of Off-Critical Zeros: Based on the positivity of the angular momentum derivative, the author argues that if a zero existed off the critical line (where $\epsilon^* > 0$), it would violate the established positivity condition. Therefore, all non-trivial zeros must lie on the critical line ($\epsilon^* = 0$).
3. Prime Spectrum: The analysis highlights a "converging spectrum of prime numbers," interpreting the zeros of $\zeta(s)$ as resonance peaks in the correlation between the weighted cosine function $\cos(\ln(x)t)$ and the fluctuation $d[J(x) - \text{Li}(x)]$.

5. Significance and Conclusion

• Alternative Path to RH: The paper offers a novel logical path to proving RH that bypasses traditional complex analysis hurdles by utilizing the Euler product directly in the critical strip and a physical "angular momentum" analogy.
• Unification of Concepts: It connects the distribution of prime numbers (via $J(x)$ and PNT) directly to the phase behavior of the zeta function, suggesting that the "spectrum of primes" dictates the location of the zeros.
• Preliminary Nature: The author explicitly states this is a "preliminary proposal" and invites scrutiny. The work relies heavily on the convergence of specific integral representations of the Euler product, which the author argues holds for $\epsilon > 0$.

Critique/Context:
While the paper presents a coherent internal logic and detailed lemmas, it is important to note that the Riemann Hypothesis remains unproven. The paper's reliance on the convergence of the Euler product in the critical strip and the specific behavior of the "angular momentum" derivative are the critical points that would require rigorous peer review and verification by the broader mathematical community. The paper is published in the "General Mathematics" (math.GM) category on arXiv, indicating it is a preprint subject to validation.