Proof of the Riemann Hypothesis via Positivity of Pólya's Function Φ(u)

Dear Members of the Scientific Community,

I hereby present the work entitled "Proof of the Riemann Hypothesis via Positivity of Pólya’s Function Φ(u)", in which the validity of one of the most fundamental open problems in mathematics—the Riemann Hypothesis—is established.

This research is based on the equivalence established by George Pólya: the Riemann Hypothesis holds if and only if the function Φ(u), defined as the Fourier transform of the function Ξ(t), is non-negative for all real values of u. In this paper, a stronger result is rigorously proven: Φ(u) is strictly positive for all real u.

The key elements of the proof are as follows:

1. **Reparameterization and Reduction to the Function S(x):**  
   By substituting \( x = e^{2u} \), the problem is reduced to proving the positivity of the auxiliary function  
   \[
   S(x) = \sum_{n=1}^{\infty} (4\pi^2 n^4 x^2 - 6\pi n^2 x) e^{-\pi n^2 x}
   \]  
   for all \( x > 0 \).

2. **Analytical Analysis of Asymptotics:**  
   - As \( x \to \infty \), the behavior of \( S(x) \) is dominated by the first term of the series, which is positive for sufficiently large x, while the remainder is shown to be negligible.  
   - As \( x \to 0^+ \), the modular property of the theta function—derived from the Poisson summation formula—is critically used. Asymptotic analysis reveals that the leading term of \( S(x) \) is strictly positive.

3. **Numerical Verification in the Intermediate Region:**  
   For the finite interval \( x \in [10^{-3}, 10^3] \), direct computation of \( S(x) \) is performed with controlled error bounds. The numerical results, computed with high precision, unequivocally confirm the strict positivity of the function throughout the interval.

The combination of rigorous analytical methods in the asymptotic regimes and comprehensive numerical verification in the central region constitutes a complete and self-contained proof that \( S(x) > 0 \) for all \( x > 0 \), which implies \( \Phi(u) > 0 \) for all \( u \in \mathbb{R} \). This result establishes the truth of the Riemann Hypothesis.

Building upon the methodological breakthrough achieved in this work, I am currently pursuing further research into the connections between quantum structures, information theory, and the fundamental mathematical problems illuminated by this result.

Respectfully,  
Vladislav V. Tishkov