  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 algebraic geometry 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 Selberg's work on
the trace formula and its comparison with the explicit formulas. The second is
based on algebraic geometry and the Riemann-Roch theorem. We establish a
framework in which one can transpose many of the ingredients of the Weil proof
as reformulated by Mattuck, Tate and Grothendieck. This framework is elaborate
and involves noncommutative geometry, Grothendieck toposes and tropical
geometry. 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 "Weil cohomology", the role of Segal's Gamma-rings
and of topological cyclic homology as a model for "absolute algebra" and as a
cohomological tool.
