Bulletin of Mathematical Sciences

, Volume 1, Issue 2, pp 245–275 | Cite as

The Möbius function and statistical mechanics

  • F. Cellarosi
  • Ya. G. Sinai
Open Access


We consider a probabilistic model for square-free numbers, and provide limit theorems for several random variables defined in our ensemble. The limit transition corresponds to the thermodynamical limit in Statistical Mechanics. We also prove some inequalities inspired by a recent conjecture by P. Sarnak concerning the randomness in the Möbius sequence, and discuss a method of summation for the Riemann zeta function ζ(s) on the vertical line \({\mathfrak{R}s = 1}\) .


Statistical Mechanic Arithmetic Progression Riemann Zeta Function Riemann Hypothesis Prime Number Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank the following people for their invaluable comments and remarks: Giovanni Gallavotti, Andrew Granville, Nicholas Katz Alex Kontorovich, Krishnan Mody, Peter Sarnak, Christopher Skinner, Domokos Szász, Anatoly M. Vershik, Ilya Vinogradov. The second author also acknowledges the financial support from NSF, Grant DMS-0600996.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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© The Author(s) 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.Landau Institute of Theoretical PhysicsRussian Academy of SciencesMoscowRussia

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