The upper bound of the Mertens function from the viewpoint of statistical mechanics

We provide some upper bounds for the Mertens function ($M(n)$: the cumulative sum of the M$\ddot{\mathrm{o}}$bius function) by an approach of statistical mechanics, in which the M$\ddot{\mathrm{o}}$bius 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 $\sqrt{\frac{B}{\alpha} n}$ ($B$ is constant) with a probability $>1-\alpha$ ($0<\alpha<1$) from the view point of the energy fluctuations in the canonical ensemble.