The Mysteries of the Real Prime

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.

The real prime	p. 1
Algebra and geometry	p. 1
The real prime	p. 3
The mystery	p. 4
The zeta function and gamma distribution	p. 6
The local zeta function	p. 6
The gamma distribution	p. 7
Remarks on global theory	p. 8
The beta distribution	p. 10
Phase space	p. 10
The beta distribution	p. 11
Special values	p. 13
Remarks on the global theory	p. 15
The p-adic hyperbolic point of view	p. 17
Chains and trees	p. 17
The p-adic gamma chain	p. 21
The p-adic symmetric beta chain	p. 24
The p-adic beta chain	p. 28
Some real hyperbolic chains	p. 31
The hyperbolic plane	p. 31
N-adic expansion	p. 31
Continued fraction expansion	p. 31
Ramanujan's garden	p. 34
The q-zeta function	p. 34
Elliptic curves	p. 36
q-series	p. 37
The q-gamma and q-beta chains	p. 44
The q-gamma chain	p. 44
The q-beta chains	p. 47
The Heisenberg relation and special basis	p. 52
The real beta chains	p. 65
The beta chain	p. 65
The Heisenberg relations and the special basis	p. 67
The real units	p. 76
Global 'chains' and higher dimensions	p. 84
Restricted direct products of chains	p. 84
Higher dimensional beta chains	p. 87
The Fourier transform	p. 98
The Tate distribution and the beta function at imaginary argument	p. 98
The Fourier--Bessel transform	p. 101
Symmetric convolution	p. 103
The basic basis and the Laguerre basis	p. 105
The beta measure	p. 110
The pure gamma basis and the cut-off basis	p. 112
The pure beta basis	p. 122
The Askey--Wilson polynomials	p. 124
The quantum group SU (1,1)	p. 129
The quantum enveloping algebra U[subscript q]	p. 129
Highest weight representation	p. 131
The Hopf algebra structure	p. 135
The universal R-matrix	p. 139
The Heisenberg group	p. 142
The Heisenberg group and its fundamental representation	p. 142
Twisted convolution and multiplication	p. 145
Matrix coefficients	p. 147
The local lattice model	p. 149
Special basis	p. 154
The global lattice model	p. 161
Automorphic forms on the Heisenberg group	p. 167
The Riemann zeta function	p. 173
The Riemann zeta function and the theta function	p. 173
The explicit sums	p. 176
The Eisenstein series connections	p. 181
The Eisenstein series and the intertwining operator	p. 188
The Riesz potential connection	p. 207
Bibliography	p. 226
Index	p. 233
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The Mysteries of the Real Prime

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