On a general density theorem

Following the pioneering work of Halász and Turán we prove a general zero-density theorem for a large class of Dirichlet series, containing the Riemann and Dedekind zeta functions. Owing to the application of an idea of Halász (contained in the above mentioned work) and a sharp Vinogradov-type estimate for the Riemann zeta function (due to Heath-Brown) the results are particularly sharp in the neighborhood of the boundary line Re $s=1$.