On the Zeros of the Riemann Zeta Function

Spectral Analysis of the Riemann Zeta Function via Adelic OperatorsThis research proposes a new way to understand the Riemann Hypothesis using the theory of operators. We create a special mathematical "ruler" (a self-adjoint scaling operator) that works within a specific space of numbers (adelic Hilbert space). By using a method called spectral regularization, we show that the famous zeros of the Riemann function correspond exactly to the points (spectrum) generated by our operator.The project focuses on two main goals: first, proving that this operator behaves predictably (has a discrete spectrum) within its domain; and second, showing that the chaotic nature of "adelic flows" ensures these zeros are distributed in a way that prevents them from repeating or falling outside the expected line. This approach connects non-commutative geometry with quantum physics, providing a clear path to proving the Riemann Hypothesis.