Ne\c{s}e Dernek, O.A.Veliev On the Riesz Basisness of the Root \ Functions of the Nonselfadjoint Sturm-Liouville Operator (27K, LATeX 2e) ABSTRACT. In this article we obtain the asymptotic formulas of arbitrary order for eigenfunctions and eigenvalues of the nonselfadjoint Sturm-Liouville operators with \ periodic and antiperiodic boundary conditions, when the potential is a summable function. Then using these asymptotic formulas, we find the conditions on Fourier coefficients of the potential for which the eigenfunctions and associated functions of these operators form a Riesz basis in $L_{2}(0,1)$.