The Number-Theoretical Spin Chain and the Riemann Zeroes
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Submission date: 30. Jun. 1997
published in: Communications in mathematical physics, 196 (1998) 3, p. 703-731
DOI number (of the published article): 10.1007/s002200050441
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It is an empirical observation that the Riemann zeta function can be well approximated in its critical strip using the Number-Theoretical Spin Chain. A proof of this would imply the Riemann Hypothesis. Here we relate that question to the one of spectral radii of a family of Markov chains. This in turn leads to the question whether certain graphs are Ramanujan.
The general idea is to explain the pseudorandom features of certain number-theoretical functions by considering them as observables of a spin chain of statistical mechanics. In an Appendix we relate the free energy of that chain to the Lewis Equation of modular theory.