Several leading mathematicians (e.g. Yuri Manin) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute algebraic geometry over a the field of one element; some like Mochizuki and Durov are thinking of a possible application of $\mathbf{F}_1$-geometry to an even stronger abc conjecture. It seems that this is one of the driving forces for studying algebraic geometry over $\mathbf{F}_1$ and that the main obstacle to materializing this proof is that the geometry over $\mathbf{F}_1$ (cf. MO what is the field with one element, applications of algebaric geometry over a field with one element) is still not satisfactorily developed. Even a longer-term attacker of the Riemann hypothesis from outside the algebraic geometry community, Alain Connes, has concentrated recently in his collaboration with Katia Consani on the development of a version of geometry over $\mathbf{F}_1$.
Could somebody outline for us the ideas in the folklore sketch of the proof of the Riemann hypothesis via absolute geometry ? Is the proof analogous to the Deligne's proof (article) of the Riemann-Weil conjecture (see wikipedia and MathOverflow question equivalent-statements-of-riemann-hypothesis-in-the-weil-conjectures) ?
Grothendieck was not happy with Deligne's proof, since he expected that the proof would/should be based on substantial progress for on motives and the standard conjectures on algebraic cycles. Is there any envisioned progress in the motivic picture based on $\mathbf{F}_1$-geometry, or even envisioned extensions of the motivic picture ?
