Thermodynamical averages for the Ising model and spectral invariants of Toeplitz matrices

We derive a general formula giving a representation of the partition function of the one-dimensional Ising model of a system of $N$ particles in the form of an explicitly defined functional of the spectral invariants of finite submatrices of a certain infinite Toeplitz matrix. We obtain an asymptotic representation of the partition function for large $N$, which can be a base for explicitly calculating some thermodynamic averages, for example, the specific free energy, in the case of a general translation-invariant spin interaction (not necessarily only between nearest neighbors). We estimate the partition function from above and below in the plane of the complex variable $\beta$ $(\beta$ is the inverse temperature) and consider the conditions under which these estimates are asymptotically equivalent as $N\to\infty$.