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MiS Preprint

The Number-Theoretical Spin Chain and the Riemann Zeroes

Andreas Knauf


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.

Jun 30, 1997
Jun 30, 1997

Related publications

1998 Repository Open Access
Andreas Knauf

The number-theoretical spin chain and the Riemann zeroes

In: Communications in mathematical physics, 196 (1998) 3, pp. 703-731