Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
21
The categories of this paper are: adap-org
13
The categories of this paper are: quant-ph
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas