A Proof to the Riemann Hypothesis Using a Simplified Xi-Function

The Riemann hypothesis has been of great interest in the mathematics community since it was proposed by Bernhard Riemann in 1859, and makes important implications about the distribution of prime numbers. We have proved the Riemann hypothesis in this paper. First, we briefly reviewed the simplified Riemann $\xi(s)$ function and its important properties. Then, through three theorems, we showed that in the critical line all zeros of the $\xi(s)$ function are simple, all local maxima are positive and all local minima are negative. Based on that, through two theorems, we applied the idea of bisection search to break up the complex domain into several sub-domains, in which the imaginary part of $\xi(s)$ takes different signs. Using the Cauchy-Riemann equations we found that the zeros of the $\xi(s)$ function cannot be located outside the critical line, and thus proved that the Riemann hypothesis is true. At the end, we gave some comments to the P\'olya's counterexample