The Number-Theoretical Spin Chain and the Riemann Zeroes

Knauf A (1998)

Publication Type: Journal article, Original article

Publication year: 1998


Publisher: Springer Verlag (Germany)

Book Volume: 196

Pages Range: 703-731

Journal Issue: 3

DOI: 10.1007/s002200050441


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.

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How to cite


Knauf, A. (1998). The Number-Theoretical Spin Chain and the Riemann Zeroes. Communications in Mathematical Physics, 196(3), 703-731.


Knauf, Andreas. "The Number-Theoretical Spin Chain and the Riemann Zeroes." Communications in Mathematical Physics 196.3 (1998): 703-731.

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