## The Mysteries of the Real PrimeIn 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. |

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### Contents

The real prime | 1 |

Global | 9 |

The beta distribution | 10 |

The padic hyperbolic point of view | 18 |

TheH | 23 |

Some real hyperbolic chains | 31 |

The fgamma and Qbeta chains | 44 |

nd higher dimensions | 84 |

oup SUi1 | 129 |

ht representation | 139 |

attice model | 167 |

ta function | 173 |

em series connections | 181 |

ein series and the intertwining operator | 188 |

otential connection | 203 |

233 | |

### Common terms and phrases

1)-regular tree action adjoint algebra associated automorphisms beta distribution beta integral beta measure boundary chain on Xp commutative constant multiple converges convolution corresponds decomposition defined denote dual eigenfunctions Eisenstein series elliptic curve equivalent explicit sum finite formula Fourier transform fundamental representation g-binom gives graph Haar measure harmonic measure heat kernel Heis(Z Heisenberg group Heisenberg relations hence highest weight Hilbert space induces isomorphism Jacobi Laguerre basis lattice model map of chains Martin kernel Mellin transform metric non-symmetric Note number operator obtain orthogonal basis p-adic and real p-adic limit parameters polynomials positive-definite function probability measure projection quotient random walk Re(s real analogue real beta chain real limit real numbers real prime respectively restricted direct product Riemann hypothesis satisfying Schrodinger model similarly subgroup subspace symmetric beta chain theory transition probabilities tree Xp U9-module unitary representation V(Qp vector zeta function