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Abstract: 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 ... (Update)
Context of citations to this paper: More
.... scattering geodesics in the modular domain [8] There exist direct relations with the works [12] 13] by Mayer, and [11] by Lanford and Ruedin, and the Riemann Hypothesis can be related to a problem concerning the spectral radius of a related Markov chain [9]. More details...
.... x Gammafi (1 1=x) since the free energy (28) of the number theoretical spin chain (see below) equals F (fi) Gamma ln( fi) fi [27]. At this point I would like to mention a relation between Farey fractions and the Riemann zeta function. The set F n of Farey fractions...
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BibTeX entry: (Update)
Knauf, A.: The Number-Theoretical Spin Chain and the Riemann Zeroes. Commun. Math. Phys. 196, 703--731 (1998) http://citeseer.ist.psu.edu/article/knauf97numbertheoretical.html More
@misc{ knauf98numbertheoretical,
author = "A. Knauf",
title = "The Number-Theoretical Spin Chain and the Riemann Zeroes",
text = "Knauf, A.: The Number-Theoretical Spin Chain and the Riemann Zeroes. Commun.
Math. Phys. 196, 703--731 (1998)",
year = "1998",
url = "citeseer.ist.psu.edu/article/knauf97numbertheoretical.html" }
Citations (may not include all citations):148 Ramanujan Graphs (context) - Lubotzky, Phillips et al. - 1988
104 Introduction to Analytic Number Theory (context) - Apostol - 1976
69 The Theory of the Riemann Zeta Function (context) - Titchmarsh - 1967
68 Explicit Group-Theoretical Constructions of Combinatorial Sc.. (context) - Margulis - 1988
36 Some applications of modular forms (context) - Sarnak - 1990
13 a Ferromagnetic Spin Chain - Knauf - 1993
13 the Thermodynamic Formalism for the Gauss Map (context) - Mayer - 1990
11 Disproof of Mertens conjecture - Odlyzko, te - 1985
9 The Phase Transition of the NumberTheoretical Spin Chain - Contucci, Knauf - 1997
9 and Invariant Measures (context) - Lubotzky - 1994
9 Formule de trace en g'eom'etrie non-commutative et hypoth `e.. (context) - Connes - 1996
8 Phases of the Number-Theoretical Spin Chain - Knauf - 1993
8 Period functions for Maass wave forms (context) - Lewis, Zagier - 1996
8 Part II: Thermodynamic Limit (context) - Knauf, On - 1994
8 Texts and Monographs in Physics (context) - Cycon, Froese et al. - 1987
8 and Statistical Mechanics (context) - Knauf - 1995
8 The Thermodynamic Formalism Approach to Selberg's Zeta Funct.. (context) - Mayer - 1991
7 Ramanujan Graphs and some Problems of Mathematical Physics (context) - Venkov, Nikitin - 1993
7 Free Energy and Correlations of the NumberTheoretical Spin C.. - Guerra, Knauf - 1996
6 Cubic Ramanujan Graphs (context) - Chiu - 1992
5 The Index of a Pair of Projections (context) - Avron, Seiler et al. - 1994
5 Spaces of holomorphic functions equivalent to the even Maass.. (context) - Lewis - 1997
5 The Low Activity Phase of Some Dirichlet Series - Contucci, Knauf - 1996
4 Ramanujan Graphs and Hecke Operators (context) - Pizer - 1990
4 The Riemann Hypothesis and Pseudorandom Features of the Mobi.. (context) - Good, Churchhouse - 1968
3 An Algorithm for Computing Modular Forms on \Gamma (context) - Pizer - 1980
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