Abstract
The paper contains an investigation of Behavior of paths of zeros of analytic theta functions. We have considered briefly an analytic representation of finite quantum systems . The analytic functions on a torus have exactly zeros. The brief introduction to the zeros of analytic functions and there time evolution is given. We have discussed the periodic finite quantum systems. we have discussed the cases when the zeros follow the same path. We have introduced the winding numbers in general. We consider the winding numbers of the zeros of analytic theta functions.