Non-vanishing modulo a prime of special values of various $L$-functions are of great importance in studying structures of relevant arithmetic objects such as class groups of number fields and Selmer groups of elliptic curves. While there have been rich results on the complex non-vanishing of the special $L$-values in a quantitative viewpoint, i.e., the number of characters with a bounded conductor giving the non-vanishing, there were few results for the residual $L$-values. In the talk, I will present several results on the residual non-vanishing of Dirichlet and modular $L$-values.
