  It is known to be difficult to find out whether a certain multivariable
function to be a characteristic function when its corresponding measure is not
tirivial to be or not to be a probability measure on R^d. Such results were not
obtained for a long while. In this paper, multidimensional polynomial Euler
product is defined as a generalization of the polynomial Euler product. By
applying the Kronecker's approximation theorem, a necessary and sufficient
condition for some polynomial Euler products to generate characteristic
functions is given. Furthermore, by using the Baker's theorem, that of some
multidimensional polynomial Euler products is also given. As one of the most
important properties of probability distributions, the infinite divisibility of
them is studied as well.
