"The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of we discuss three strategies, working concretely at the level of the explicit formulas. The first strategy is “analytic” and is based on Riemannian spaces and and its comparison with the explicit formulas. The second is based on algebraic geometry and the . We establish a framework in which one can transpose many of the ingredients of the Weil proof as reformulated by , Tate and Grothendieck. This framework is elaborate and involves noncommutative geometry, es and . We point out the remaining difficulties and show that RH gives a strong motivation to develop algebraic geometry in the emerging world of characteristic one. Finally we briefly discuss a third strategy based on the development of a suitable “”, the role of Segal’s Γ-rings and of topological cyclic homology as a model for “absolute algebra” and as a cohomological tool."