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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 ?

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# Riemann conjecturehypothesis via absolute geometry

Several leading mathematician mathematicians (e.g. Yuri Manin) wrote have written or said publicly that there is a known outline of a likely natural proof of the Riemann conjecture hypothesis using the absolute algebraic geometry over a field of one element; some like Mochizuki and Durov think 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 driven force to study the driving forces for studying algebraic geometry over $\mathbf{F}_1$ and that the main obstacle to materialize 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 some long-term attackers on a longer-term attacker of the Riemann hypothesis from non-algebraic outside the algebraic geometry communitylike , Alain Connes, has concentrated recently in his collaboration with Katia Consani to on the development of a version of the geometry over $\mathbf{F}_1$.

Could somebody outline for us the ideas in folk the folklore sketch of the proof of the Riemann conjecture hypothesis via the 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 the Deligne's proofas , since he expected that the proof would/should be based on the substantial progress on for motives and the standard conjectures. Is there an any envisioned progress in the motivic picture based on $\mathbf{F}_1$-geometry, or even envisioned extensions of the motivic picture ?

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