**Title:**Ideas for the Riemann Hypothesis

**Abstract:**In 1973, Bombieri gave a new proof of Weil’s theorem, the Riemann Hypothesis for the zeta function of a curve defined over a finite field. We will give a historic introduction to the Riemann hypothesis in number theory, and explain the analogous situation for curves. Bombieri’s proof has two parts. The first step is an upper bound for the point counting function and is the most interesting for the Riemann Hypothesis. The second step involves the Galois cover of the curve, and gives ideas for the Generalized Riemann Hypothesis.