The Riemann Hypothesis (RH) stands as one of the most profound and enduring problems in mathematics, asserting that all nontrivial zeros of the Riemann zeta function lie on the critical line ℜ(s)=12ℜ(s)=21. This conjecture, deeply intertwined with the distribution of prime numbers, has guided much of modern analytic number theory. Despite significant computational and theoretical advances, RH remains unproven. This article surveys the latest analytic number theory approaches to RH, including new zero-free regions, refinements in the study of Dirichlet series, and connections with the growth of arithmetic functions. We also discuss recent breakthroughs, computational verifications, and the broader implications of RH for mathematics.