Communications in Mathematical Physics

, Volume 196, Issue 3, pp 703–731

The Number-Theoretical Spin Chain and the Riemann Zeroes

  • Andreas Knauf

DOI: 10.1007/s002200050441

Cite this article as:
Knauf, A. Comm Math Phys (1998) 196: 703. 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Andreas Knauf
    • 1
  1. 1.Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22–26, D-04103 Leipzig, Germany.¶E-mail: knauf@mis.mpg.deDE