Weil's Riemann Hypothesis is a deep result that I don't fully understand, but it has understandable corollaries which interest me. For example:

(a) For any projective curve $X$ satisfying certain conditions, the number $N$ of points in $X$ with coordinates in $\mathrm{GF}(q)$ satisfies $|N-(q+1)|\leq\mathrm{const}\cdot\sqrt{q}$. (The deviation is $0$ when $X$ is a projective line.)

(b) For any nontrivial multiplicative character $\chi$ on $\mathrm{GF}(q)$ and any polynomial $f$ of degree $n$ satisfying certain conditions, we have

$$\bigg|\sum_{x\in\mathrm{GF}(q)}\chi(f(x))\bigg|\leq(n-1)\sqrt{q}.$$

Questions:

Is there a reference (legible to an English-speaking non-expert in the field) which gives the rigorous statements of these corollaries? In particular, I would like conditions which one can verify without a background in algebraic geometry.

Are there other corollaries of Weil's Riemann Hypothesis which are also widely understandable?

**EDIT:**I'm mostly interested in the Riemann Hypothesis, but I'm also happy to learn understandable consequences of the other Weil conjectures and related results.

Riemann hypothesisfor curves, or in any aspects of the Weil conjectures for curves (of which RH is the hardest part, but also includes rationality and a functional equation for the zeta-function of the curve)? What about for higher-dimensional varieties? I ask because one of the answers below is an application of the rationality of the zeta-function for higher-dimensional varieties, not involving RH for that zeta-function. Is that a kind of consequence you're interested in too? It'd be good to clarify this in your question. $\endgroup$ – KConrad Jul 21 '14 at 14:35