An additive type problem for the Fourier coefficients of automorphic cusp forms and its application to the zeros of the associated $L$-functions on the critical line

In our talk we give a non-trivial estimate for the analogue of the additive Dirichlet divisor problem for the Fourier coefficients coming from an automorphic cusp form for the group $\Gamma_0(D)$ with an arbitrary character. As a result, we obtain that a positive proportion of non-trivial zeros are on the critical line for $L$-functions associated with such automorphic forms.