Functional integration, zeta regularization and correlators of third components of spins in the $XXO$-Heisenberg model

New representation for the generating function of static correlators of third components of spins in the $XXO$-Heisenberg model is advanced in the form of Gaussian functional integral. The generating function, as well as the partition function of the model, are calculated by means of zeta regularization procedure. Formulas for certain correlation functions at nonzero temperature are obtained.