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An Essay on the Riemann Hypothesis

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Abstract

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 Γ-rings and of topological cyclic homology as a model for “absolute algebra” and as a cohomological tool.

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Notes

  1. 1.

    Similar counting functions were already present in Chebyshev’s work.

  2. 2.

    More precisely Riemann writes \(\sum _{\mathfrak{R}(\alpha )>0}\left (\mathrm{Li}(x^{\frac{1} {2} +\alpha i}) +\mathrm{ Li}(x^{\frac{1} {2} -\alpha i})\right )\) instead of \(\sum _{\rho }\mathrm{Li}(x^{\rho })\) using the symmetry ρ → 1 −ρ provided by the functional equation, to perform the summation.

  3. 3.

    See [44, Chap. VII] for detailed support to Selberg’s comment.

  4. 4.

    My warmest thanks to Michael Th. Rassias for the communication.

  5. 5.

    One of the topics in which John Nash made fundamental contributions.

  6. 6.

    And at which the value of F(u) is defined as the average of the right and left limits there.

  7. 7.

    Where the argument of r n is either 0 or \(-\pi /2\).

  8. 8.

    I am grateful to S. Gaubert for pointing out this early occurrence.

  9. 9.

    As mentioned above, this result was obtained already for matrices in 1962 by R. Cuninghame-Green.

  10. 10.

    As seen when using \(\mathbb{R}_{\mathrm{max}}\) as the target of a valuation.

  11. 11.

    Where 0 is the base point.

  12. 12.

    Equivalently \(\mathbb{S}\)-algebras.

  13. 13.

    A measurable group homomorphism from \(\mathbb{Z}_{p}^{\times }\) to \(\mathbb{C}^{\times }\) cannot be injective.

  14. 14.

    In 2012 I had to give, in the French academy of Sciences, the talk devoted to the 200th anniversary of the birth of Evariste Galois. On that occasion I read for the n + 1th time the book of his collected works and was struck by the pertinence of the above quote in the analogy with L-functions. In the case of function fields one is dealing with Weil numbers and one knows a lot on their Galois theory using results such as those of Honda and Tate cf. [94].

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Acknowledgements

I am grateful to J. B. Bost for the reference [94], to J. B. Bost, P. Cartier, C. Consani, D. Goss, H. Moscovici, M. Th. Rassias, C. Skau and W. van Suijlekom for their detailed comments, to S. Gaubert for his help in Sect. 4.2 and to Lars Hesselholt for his comments and for allowing me to mention his forthcoming paper [64].

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  1. Collège de France, 3 Rue d’Ulm, Paris, France

    Alain Connes

  2. IHÉS, 35 Route de Chartres, Bures-sur-Yvette, France

    Alain Connes

  3. Ohio State University, Columbus, OH, USA

    Alain Connes

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  1. Department of Mathematics, Senior Research Mathematician, Princeton University, Princeton, New Jersey, USA

    John Forbes Nash, Jr.

  2. Department of Mathematics, Princeton University, Princeton, New Jersey, USA

    Michael Th. Rassias

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Connes, A. (2016). An Essay on the Riemann Hypothesis. In: Nash, Jr., J., Rassias, M. (eds) Open Problems in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-32162-2_5

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