The Riemann zeta function, symbol of the convergence between complex analysis and number theory, occupies a central place in modern mathematics. Since its introduction by Bernhard Riemann in 1859, this function has established itself as an essential tool for understanding the distribution of prime numbers. This work is distinguished by the ambition to propose a formal proof of the Riemann Hypothesis (HR), a conjecture which intimately links the non-trivial zeros of ��(��) to the structure of integers. By exploring its analytical, geometric and arithmetic aspects, this text constitutes a significant step forward in the quest for this proof. The objective of this work is twofold: on the one hand, to offer an in-depth presentation of the fundamental properties of ��(��), such as its functional equation, its Euler product, and the symmetry of its zeros; on the other hand, rigorously demonstrate that all non-trivial zeros of ��(��) lie on the critical line ℜ(��)=1/2. In this way, it is aimed not only at specialists but also at mathematics enthusiasts, by offering them a unique and detailed perspective on one of the greatest mathematical mysteries. Through a rigorous methodology, illustrated by demonstrations by contradiction and analytical visualizations, this text highlights the profound consequences of HR, not only on number theory, but also on other branches of mathematics. It invites the reader to explore this hidden harmony that links the analytical properties of complex functions and the regularity of prime numbers.