Almost non-degenerate functions and a Zariski pair of links

Let $f(\mathbf{z})$ be an analytic function defined in the neighborhood of the origin of ${\mathbb{C}}^n$ which have some Newton degenerate faces. We generalize the Varchenko formula for the zeta function of the Milnor fibration of a Newton non-degenerate function $f$ to this case. As an application, we give an example of a pair of hypersurfaces with the same Newton boundary and the same zeta function with different tangent cones.