The Mysteries of the Real Prime

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