Warning: I am not an expert here but I'll give this a shot.
In the analogy between number fields and function field, Riemmann's Riemann's zeta funnction is the $\zeta$ function for $\mathrm{Spec} \ \mathbb{Z}$. Note that $\mathrm{Spec} \ \mathbb{Z}$ is one dimensional. So proving the Riemmann Riemann hypothesis should be like proving the Weil conjectures for a curve, which was done by Weil. Deligne's achievement was to prove the Weil conjectures for higher dimensional varieties which, according to this analogy, should be less relevant.
I wrote a blog post about one of the standard ways to prove the Riemann hypothesis for a curve $X$ (over $\mathbb{F}_p$). Note that a central role is played by the surface $X \times X$. I believe the $\mathbb{F}_1$ approach is to invent some object which can be called $(\mathrm{Spec} \ \mathbb{Z}) \times_{\mathbb{F}_1} (\mathrm{Spec} \ \mathbb{Z})$.
