2018/07/10 by Rong Qiang Wei, Wei, Rong Qiang
Mathematics · #FOS: Mathematics #General Mathematics (math.GM) #math.GM
paper · pdf · doi:10.48550/arxiv.1807.09085
6 pages
arxiv created 2019/08/25 · arxiv updated 2019/08/27
We provide some upper bounds for the Mertens function (M(n): the cumulative sum of the Mobius function) by an approach of statistical mechanics, in which the Mobius function is taken as a particular state of a modified one-dimensional (1D) Ising model without the exchange interaction between the spins. Further, based on the assumptions and conclusions of the statistical mechanics, we discuss the problem that M(n) can be equivalent to the sum of an independent random sequence. It holds in the sense of equivalent probability, from which another two upper bounds for the M(n) can be inferred. Besides, if M(n) is a measured quantity, its upper bound is √\fracBα n (B is constant) with a probability >1-α (0<α<1) from the view point of the energy fluctuations in the canonical ensemble.