Geometric (Pythagorean) and Functional Analysis in Support of the Riemann Hypothesis: A Number-Theoretic Foundation with a New Spatial Interpretation

We propose a geometric-functional hypothesis supporting the Riemann Hypothesis, grounded in number theory and inspired by the Pythagorean theorem. By treating the complex argument of the Riemann zeta function as a vector in the complex plane, we analyze the modulus of the function in relation to the real and imaginary parts of its input. We argue that only the critical line ℜ(s) = 1/2 yields a balanced vector structure that satisfies both the Pythagorean identity and the necessary conditions for ζ(s) to vanish. Additional reasoning involving vector alignment between known nontrivial zeros and geometric constraints supports the uniqueness of the critical line. The work represents a collaborative exploration between a human researcher and an artificial intelligence (ChatGPT), highlighting a novel approach to one of mathematics’ most profound unsolved problems.