The prevalent theories for polyelectrolytes have not been able to describe the experimentally observed ion specificity found at high ion concentrations for counterion condensation and conformational transitions. One important source of ion specificity missed in these theories is the dispersion force that acts between an ion and the cylindrical polyion surface. Here, we solve the cylindrical Poisson-Boltzmann equation numerically in the presence of ionic dispersion potentials. We demonstrate that ionic dispersion forces have important influences on ionic distributions and potentials. This influence increases with increasing concentration reflecting that specific ion effects are observed only at high concentrations.