In this talk I will make first an introduction to the classical steepest descent method by deducing the asymptotics of the Airy functions. The rest of the talk contains an application of the Riemann-Hilbertproblem for orthogonal polynomials associated to the generalized Jacobi weight in the study of the local zero behavior of orthogonal polynomials around an algebraic singularity. It will be shown that the so-called fine zero spacing unravels in the general case, and the asymptotic behavior of neighbouring zeros around the singularity can be described with the zeros of the function linear combination of Bessel functions of the first kind. Moreover, by using Sturm-Liouville theory, I will show the behavior of this linear combination of Bessel functions, thus providing estimates for the zeros in question. The talk is based on a paper (published in Constructive Approximation 47 (2018) 407-435)with Tivadar Danka (University of Szeged, Hungary) and an essential part of the study comes from the Riemann-Hilbert problem for the orthogonal polynomial associated to the generalized Jacobi weightwhich has been considered by Marteen Vanlessen, who used the non linear steepest descent method discovered by Percy Deift and Xin Zhou.