The author [Math. Pannonica (N. S.) 29, No. 2, 238–243 (2023; Zbl 07794426)] proved a general zero density theorem for a large class of functions including the Riemann zeta function, Dedekind zeta functions and Dirichlet \(L\)-functions. In the present paper a (slight) improvement of that general theorem is given. The proof does not need any deep result concerning the zeta function or Turán’s power sum method, only a simple but ingenious idea of G. Halász [Acta Math. Acad. Sci. Hung. 19, 365–403 (1968; Zbl 0165.05804)].