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基本説明
Highly topical and original monograph, introducing the authors work on the Riemann zeta function and its adelic interpretation of interest to a wide range of mathematicians and physicists.
Full Description
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.
Contents
0. Introduction ; 1. The Real Prime ; 2. The Zeta function and Gamma distribution ; 3. The Beta distribution ; 4. The p-adic hyperbolic point of view ; 5. Some real hyberbolic chains ; 6. Ramanujan's Garden ; 7. The q-Gamma and q-Beta chains ; 8. The real Beta chains ; 9. Global "chains" and higher dimensions ; 10. The Fourier transform ; 11. The quantum group ; 12. The Heisenberg group ; 13. The Riemann Zeta function ; 14. References / Bibliography
