The Riemann Zeta Function and Its Zeroes

This dissertation includes a detailed of the Riemann zeta functions; with a particular focus its analytic continuation, functional equation and application. We will start with the historical background. Following this we cover certain important preliminaries which are needed to use the functional equation. We then define the Riemann zeta function and prove the functional equation. In addition to this that, we show the Riemann zeta function has generalization in form of the Dirichlet L-function. Then, the zeroes of the Riemann zeta function will be studied. Finally, we establish the zero free region of Riemann zeta function.