Abstract
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 \({\mathfrak{R}s = 1}\) .
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Acknowledgments
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.
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Cellarosi, F., Sinai, Y.G. The Möbius function and statistical mechanics. Bull. Math. Sci. 1, 245–275 (2011). https://doi.org/10.1007/s13373-011-0011-6
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DOI: https://doi.org/10.1007/s13373-011-0011-6