An essay on the Riemann Hypothesis

@article{Alain2015AnEO,
  title={An essay on the Riemann Hypothesis},
  author={Connes Alain},
  journal={arXiv: Number Theory},
  year={2015}
}
  • Connes Alain
  • Published 18 September 2015
  • Mathematics
  • arXiv: Number Theory
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… 
Noncommutative Geometry, the Spectral Standpoint
  • A. Connes
  • Mathematics
    New Spaces in Physics
  • 2019
We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative
BC-system, absolute cyclotomy and the quantized calculus
We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the ”zeta sector” of the latter space as the Scaling Site. The new
ON RIEMANN ZEROS AND WEIL CONJECTURES
. The article aims to motivate the study of the relations between the Riemann zeros, and the zeros of the Weil polynomial of a hyper-elliptic curve over finite fields, beyond the well-known formal
From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis
The novelty of the Jean Pierre Badiali last scientific works stems to a quantum approach based on both (i) a return to the notion of trajectories (Feynman paths) and (ii) an irreversibility of the
On Prime Numbers and The Riemann Zeros
. The current research regarding the Riemann zeros suggests the existence of a non-trivial algebraic/analytic structure on the set of Riemann zeros [1, 2, 3]. The duality between primes and Riemann
Weil positivity and trace formula the archimedean place
We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula of Connes (Sel Math (NS) 5(1):29–106, 1999). We
The Scaling Hamiltonian
We first explain the link between the Berry-Keating Hamiltonian and the spectral realization of zeros of the Riemann zeta function of the first author, and why there is no conflict at the
All Complex Zeros of the Riemann Zeta Function Are on the Critical Line: Two Proofs of the Riemann Hypothesis
I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeroes of the Riemann Zeta
On The Complex Zeros of The Riemann Zeta Function
A mathematical proof is only true if the proof can be reproducible, and perhaps by alternative means than that employed in the first proof. A proof of the Riemann Hypothesis should be generalizable
The Riemann Zeros as Spectrum and the Riemann Hypothesis
TLDR
A spectral realization of the Riemann zeros based on the propagation of a massless Dirac fermion in a region of Rindler spacetime and under the action of delta function potentials localized on the square free integers suggests a proof of theRiemann hypothesis in the limit where the potentials vanish.
...
...

References

SHOWING 1-10 OF 139 REFERENCES
THE WITT CONSTRUCTION IN CHARACTERISTIC ONE AND QUANTIZATION
We develop the analogue of the Witt construction in characteristic one. We construct a functor from pairs (R; ) of a perfect semi-ring R of characteristic one and an element > 1 of R to real Banach
Noncommutative geometry, arithmetic, and related topics : proceedings of the Twenty-first Meeting of the Japan-U.S. Mathematics Institute
This valuable collection of essays by some of the world's leading scholars in mathematics presents innovative and field-defining work at the intersection of noncommutative geometry and number theory.
On the topological cyclic homology of the algebraic closure of a local field
The cyclotomic trace provides a comparison of the algebraic K-theory spectrum and a pro-spectrum TR that is built from the cyclic fixed points of topological Hochschild homology. In a previous paper
Noncommutative Geometry
Noncommutative Spaces It was noticed a long time ago that various properties of sets of points can be restated in terms of properties of certain commutative rings of functions over those sets. In
Sheaves in geometry and logic: a first introduction to topos theory
This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various
Tropical Mathematics
This article is based on the Clay Mathematics Senior Scholar Lecture that was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture was the tropical approach in
The Explicit Formula and the conductor operator
I give a new derivation of the Explicit Formula for an arbitrary number field and abelian Dirichlet-Hecke character, which treats all primes in exactly the same way, whether they are discrete or
From the sixteenth Hilbert problem to tropical geometry
Abstract.Hilbert’s problem on the topology of algebraic curves and surfaces (the sixteenth problem from the famous list presented at the second International Congress of Mathematicians in 1900) was
...
...