Preprint 15/1997

The Number-Theoretical Spin Chain and the Riemann Zeroes

Andreas Knauf

Contact the author: Please use for correspondence this email.
Submission date: 30. Jun. 1997
Pages: 37
published in: Communications in mathematical physics, 196 (1998) 3, p. 703-731 
DOI number (of the published article): 10.1007/s002200050441
Download full preprint: PDF (500 kB), PS ziped (321 kB)

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.

24.02.2017, 01:40