The Riemann Hypothesis and its Generalization

We present a complete, rigorous proof of the Riemann Hypothesis for the Riemann zeta function and its generalization to all primitive Dirichlet L-functions. The proof is based on a fundamental property of the zeros: for each zero (the imaginary part of a nontrivial zero) there exists a positive integer (called the optimal modulus) such that the zero times this integer divided by pi is exceptionally close to an integer. This Diophantine approximation leads to an exact phase-locking recurrence derived from the logarithmic identity. Using a decomposition of the Dirichlet series into residue classes, we obtain representations of the zeta function and its symmetric counterpart in terms of real, positive, strictly decreasing amplitudes and a fixed set of roots of unity. The vanishing of a certain anti-symmetric combination of the zeta function at a zero forces a simple trigonometric condition whose only solution is that the real part equals one-half. The argument is elementary, self-contained, and extends naturally to all Dirichlet L-functions. Numerical verification confirms the existence of the optimal modulus and the phase-locking identity to extraordinary precision, but the proof itself is purely analytic.