  Number theory is considered, by proposing quantum mechanical models and
string-like models at zero and finite temperatures, where the factorization of
number into prime numbers is viewed as the decay of particle into elementary
particles conserving energy. In these models, energy of a particle labeled by
an integer $n$ is assumed or derived to being proportional to $\ln n$. The
one-loop vacuum amplitudes, the free energies and the partition functions at
finite temperature of the string-like models are estimated and compared with
the zeta functions. The $SL(2, {\bf Z})$ modular symmetry, being manifest in
the free energies is broken down to the additive symmetry of integers, ${\bf
Z}_{+}$, after interactions are turned on. In the dynamical model existing
behind the zeta function, prepared are the fields labeled by prime numbers. On
the other hand the fields in our models are labeled, not by prime numbers but
by integers. Nevertheless, we can understand whether a number is prime or not
prime by the decay rate, namely by the corresponding particle can decay or can
not decay through interactions conserving energy. Among the models proposed,
the supersymmetric string-like model has the merit of that the zero point
energies are cancelled and the energy levels may be stable against radiative
corrections.
