An Essay on the Riemann Hypothesis

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.

[1]  A. Connes,et al.  Schemes over 𝔽1 and zeta functions , 2009, Compositio Mathematica.

[2]  The Explicit Formula and the conductor operator , 1999, math/9902080.

[3]  O. Viro.  On basic concepts of tropical geometry , 2011 .

[4]  Serguei Norine,et al.  Riemann–Roch and Abel–Jacobi theory on a finite graph , 2006, math/0608360.

[5]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[6]  Stéphane Gaubert,et al.  Methods and Applications of (MAX, +) Linear Algebra , 1997, STACS.

[7]  A. Connes.  THE WITT CONSTRUCTION IN CHARACTERISTIC ONE AND QUANTIZATION , 2010, 1009.1769.

[8]  A. Connes,et al.  The Arithmetic Site , 2014, 1405.4527.

[9]  R. Schoof,et al.  Number fields and function fields: two parallel worlds , 2005 .

[10]  Alain Connes,et al.  The hyperring of adèle classes , 2011 .

[11]  Ib Madsen,et al.  On the K-theory of finite algebras over witt vectors of perfect fields , 1997 .

[12]  G. Mikhalkin.  Enumerative tropical algebraic geometry , 2003 .

[13]  L. Illusie.  Complexe de de Rham-Witt et cohomologie cristalline , 1979 .

[14]  N. Durov.  New Approach to Arakelov Geometry , 2007, 0704.2030.

[15]  J. Arthur.  An Introduction to the Trace Formula , 2005 .

[16]  A. Weil,et al.  Sur les “formules explicites” de la théorie des nombres premiers , 1979 .

[17]  Jacques Tits,et al.  Sur les analogues algébriques des groupes semi-simples complexes , 1957 .

[18]  Alain Connes,et al.  Trace formula in noncommutative geometry and the zeros of the Riemann zeta function , 1998, math/9811068.

[19]  Grigory Mikhalkin,et al.  Tropical curves, their Jacobians and Theta functions , 2006 .

[20]  P. Deligne.  La conjecture de Weil. I , 1974 .

[21]  László Lovász,et al.  Chip-firing Games on Graphs , 1991, Eur. J. Comb..

[22]  R. Steinberg,et al.  A geometric approach to the representations of the full linear group over a Galois field , 1951 .

[23]  Jeffrey C. Lagarias,et al.  Complements to Li's Criterion for the Riemann Hypothesis☆ , 1999 .

[24]  Jorge V. José,et al.  Chaos in classical and quantum mechanics , 1990 .

[25]  B. Riemann,et al.  Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse , 2013 .

[26]  A. Connes,et al.  Cyclic homology, Serre's local factors and the -operations , 2012, 1211.4239.

[27]  A. Connes,et al.  Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory , 1995 .

[28]  The Scaling Site , 2015, 1507.05818.

[29]  Shlomo Sternberg,et al.  Geometric Asymptotics, Revised edition , 1977 .

[30]  André Weil,et al.  Œuvres scientifiques : collected papers , 1979 .

[31]  B. Julia.  Statistical Theory of Numbers , 1990 .

[32]  A. Connes,et al.  Geometry of the Arithmetic Site , 2015, 1502.05580.

[33]  Imre Simon,et al.  Limited subsets of a free monoid , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[34]  A. Connes,et al.  On the arithmetic of the BC-system , 2011, 1103.4672.

[35]  A. Connes,et al.  Noncommutative geometry and motives: The thermodynamics of endomotives , 2005, math/0512138.

[36]  E. Bombieri.  The Classical Theory of Zeta and L-Functions , 2010 .

[37]  Michael Kerber,et al.  A Riemann–Roch theorem in tropical geometry , 2006, math/0612129.

[38]  J. Dixmier,et al.  On some C∗-algebras considered by Glimm , 1967 .

[39]  Douglas Lind,et al.  Non-archimedean amoebas and tropical varieties , 2004, math/0408311.

[40]  N. Kurokawa.  Multiple Zeta Functions: An Example , 1992 .

[41]  D. Hejhal.  The selberg trace formula and the riemann zeta function , 1976 .

[42]  A. Deitmar.  schemes and toric varieties. , 2008 .

[43]  N. Jacobson.  The Radical and Semi-Simplicity for Arbitrary Rings , 1945 .

[44]  Oleg Viro,et al.  From the sixteenth Hilbert problem to tropical geometry , 2008 .

[45]  C. Deninger.  On the Γ-factors attached to motives , 1991 .

[46]  F. Bruhat.  Distributions sur un groupe localement compact et applications à l’étude des représentations des groupes $p$-adiques , 1961 .

[47]  P. Cartier,et al.  Une nouvelle interprétation de la formule des traces de Selberg , 2007 .

[48]  Shai Haran,et al.  Riesz potentials and explicit sums in arithmetic , 1990 .

[49]  Xian-jin Li,et al.  The Positivity of a Sequence of Numbers and the Riemann Hypothesis , 1997 .

[50]  C. Weibel.  The Hodge ?ltration and cyclic homology , 1997 .

[51]  A. Deitmar.  Schemes over \( \mathbb{F}_1 \) , 2005 .

[52]  G. Litvinov.  Tropical Mathematics, Idempotent Analysis, Classical Mechanics and Geometry , 2010, 1005.1247.

[53]  A. Grothendieck.  Sur une note de Mattuck-Tate. , 1958 .

[54]  A. Weil.  Sur la Théorie du Corps de Classes , 1951 .

[55]  Harold M. Edwards,et al.  Riemann's Zeta Function , 1974 .

[56]  L. Hesselholt.  On thep-typical curves in Quillen'sK-theory , 1996 .

[57]  L. Hesselholt.  Periodic topological cyclic homology and the Hasse-Weil zeta function , 2016 .

[58]  A. Connes,et al.  The universal thickening of the eld of real numbers , 2012 .

[59]  R. Nikoukhah,et al.  Convex analysis and spectral analysis of timed event graphs , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[60]  V. Guillemin.  Lectures on spectral theory of elliptic operators , 1977 .

[61]  C. Weibel.  Cyclic homology for schemes , 1996 .

[62]  Jean-Pierre Serre.  Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures) , 1970 .

[63]  V. Kolokoltsov,et al.  Idempotent Analysis and Its Applications , 1997 .

[64]  M. Atiyah,et al.  A Lefschetz Fixed Point Formula for Elliptic Complexes: I , 1967 .

[65]  C. Soulé.  Les vari\'et\'es sur le corps \`a un \'el\'ement , 2003, math/0304444.

[66]  J. Tate.  Fourier analysis in number fields and Hecke's zeta-functions , 1950 .

[67]  M. Lydakis.  Smash products and Γ-spaces , 1999, Mathematical Proceedings of the Cambridge Philosophical Society.

[68]  Dhar,et al.  Self-organized critical state of sandpile automaton models. , 1990, Physical review letters.

[69]  M. Gutzwiller.  Classical Quantization of a Hamiltonian with Ergodic Behavior , 1980 .

[70]  A. Connes,et al.  Absolute algebra and Segal's Γ-rings: Au dessous de Spec(Z)‾ , 2016 .

[71]  P. Deligne.  Valeurs de fonctions L et périodes d’intégrales , 1979 .

[72]  S. Lane,et al.  Sheaves In Geometry And Logic , 1992 .

[73]  Alexander Postnikov,et al.  Trees, parking functions, syzygies, and deformations of monomial ideals , 2003 .

[74]  A. Selberg.  Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series , 1956 .

[75]  Bjørn Ian Dundas,et al.  The Local Structure of Algebraic K-Theory , 2012 .