In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes
American Mathematical Soc., 2008 - 558 pages
Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possible--or, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line. In this book, the author proposes a new approach to understand and possibly solve the Riemann Hypothesis. His reformulation builds upon earlier (joint) work on complex fractal dimensions and the vibrations of fractal strings, combined with string theory and noncommutative geometry. Accordingly, it relies on the new notion of a fractal membrane or quantized fractal string, along with the modular flow on the associated moduli space of fractal membranes. Conjecturally, under the action of the modular flow, the spacetime geometries become increasingly symmetric and crystal-like, hence, arithmetic. Correspondingly, the zeros of the associated zeta functions eventually condense onto the critical line, towards which they are attracted, thereby explaining why the Riemann Hypothesis must be true. Written with a diverse audience in mind, this unique book is suitable for graduate students, experts and nonexperts alike, with an interest in number theory, analysis, dynamical systems, arithmetic, fractal or noncommutative geometry, and mathematical or theoretical physics.
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String Theory on a Circle and TDuality Analogy with the Riemann Zeta Function
Fractal Strings and Fractal Membranes
Noncommutative Models of Fractal Strings Fractal Membranes and Beyond
Towards an Arithmetic Site Moduli Spaces Fractal Strings and Membranes
The Weil Conjectures and the Riemann Hypothesis
The Poisson Summation Formula with Applications
analogy Appendix arithmetic geometries associated asymptotic automorphic C*-algebra Chapter commutative compact complex dimensions Conjecture Connes context corresponding counterpart defined definition denotes Dirichlet discussion dual duality dynamical equivalently Euler product example extended factor field theory finite fields flow of zeta fractal membranes fractal strings functional equation Furthermore given groupoid hence Hilbert space integers L-functions L-series Lap-vF2 LapNel lattice manifold mathematical metric modular flow modular forms moduli space Moreover multiplicative noncommutative geometry noncommutative space notation notion number fields number theory Open Problem particular partition function Penrose tilings physical Poisson Summation Formula poles precisely prime membrane Prime Number quantization quasicrystals Recall Remark renormalization resp Ricci flow Riemann Hypothesis Riemann zeta function Riemannian satisfies Section Selberg Class self-dual self-similar self-similar membrane self-similar string sequence space of fractal spacetime spectral triple string theory subset suitable T-duality Theorem topology torus vertex algebra Zc(s zeros
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