The Mysteries of the Real Prime (London Mathematical Society Monographs, New Series)

In this important and original monograph, useful for both academic and professional researchers and students of mathematics and physics, the author describes his work on the Riemann zeta function and its adelic interpretation. It provides an original point of view, bringing new, highly useful dictionaries between different fields of mathematics. It develops an arithmetical approach to the continuum of real numbers and unifies many areas of mathematics including: Markov Chains, q-series, Elliptic curves, the Heisenberg group, quantum groups, and special functions (such as the Gamma, Beta, Zeta, theta, Bessel functions, the Askey-Wilson and the classical orthagonal polynomials) The text discusses real numbers from a p-adic point of view, first mooted by Araeklov. It includes original work on coherent theory, with implications for number theory and uses ideas from probability theory including Markov chains and noncommutative geometry which unifies the p-adic theory and the real theory by constructing a theory of quantum orthagonal polynomials. Title page Preface 1 The real prime 1.1 Algebra and geometry 1.2 The real prime 1.3 The mystery 2 The zeta function and gamma distribution 2.1 The local zeta function 2.2 The gamma distribution 2.3 Remarks on global theory 3 The beta distribution 3.1 Phase space 3.2 The beta distribution 3.3 Special values 3.4 Remarks on the global theory 4 The p-adic hyperbolic point of view 4.1 Chains and trees 4.2 The p-adic gamma chain 4.3 The p-adic symmetric beta chain 4.4 The p-adic beta chain 5 Some real hyperbolic chains 5.1 The hyperbolic plane 5.2 N-adic expansion 5.3 Continued fraction expansion 6 Ramanujan's garden 6.1 The q-zeta function 6.2 Elliptic curves 6.3 q-series 7 The q-gamma and q-beta chains 7.1 The q-gamma chain 7.2 The q-beta chains 7.3 The Heisenberg relation and special basis 8 The real beta chains 8.1 The beta chain 8.2 The Heisenberg relations and the special basis 8.3 The real units 9 Global 'chains' and higher dimensions 9.1 Restricted direct products of chains 9.2 Higher dimensional beta chains 10 The Fourier transform 10.1 The Tate dîstribution and the beta function at imaginary argument 10.2 The Fourier-Bessel transform 10.3 Symmetric convolution 10.4 The basic basis and the Laguerre basis 10.5 The beta measure 10.6 The pure gamma basis and the cut-off basis 10.7 The pure beta basis 10.8 The Askey-Wilson polynomials 11 The quantum group SU(l,l) 11.1 The quantum enveloping algebra U_q 11.2 Highest weight representation 11.3 The Hopf algebra structure 11.4 The universal R-matrix 12 The Heisenberg group 12.1 The Heisenberg group and its fundamental representation 12.2 Twisted convolution and multiplication 12.3 Matrix coefficients 12.4 The local lattice model 12.5 Special basis 12.6 The global lattice model 12.7 Automorphic forms on the Heisenberg group 13 The Riemann zeta function 13.1 The Riemann zeta fonction and the theta function 13.2 The explicit sums 13.3 The Eisenstein series connections 13.4 The Eisenstein series and the intertwining operator 13.5 The Riesz potential connection Bibliography Index The arithmetic of number fields, which are finite extensions of the field of rational numbers Q, resembles the geometric theory of function fields, the fields of meromorphic functions on the one-dimensional objects of geometry-the curves. Highly topical and original monograph, introducing the author's work on the Riemann zeta function and its adelic interpretation of interest to a wide range of mathematicians and physicists.