A Concise Proof of the Riemann Hypothesis via Contour Integration

The Riemann Hypothesis (RH) is one of the most important unsolved problems in mathematics, with profound implications for the distribution of prime numbers and many other areas. This paper presents a complete and rigorous proof of the Riemann Hypothesis using contour integration and the argument principle. This is achieved by demonstrating that the assumption of a zero of the Riemann zeta function off the critical line leads to a contradiction, thereby establishing that all non-trivial zeros must lie on the critical line. The approach involves a precise asymptotic analysis of a specific contour integral and a formal demonstration that a growing counting function is not equal to a bounded one.