A general mean-value theorem for multiplicative functions taking values in the unit disc was given by Wirsing (1967) and Halász (1968). We consider a multiplicative function f belonging to a certain class of arithmetical functions and let F(s) be the associated Dirichlet series. In this setting, we obtain new Halász-type results for the logarithmic mean value of f. More precisely, we give estimates in terms of the size of $|F(1+1/\\log x)|$ and show that these estimates are sharp.  

As a consequence, we obtain a non-trivial zero-free region for partial sums of L-functions belonging to our class. We also report on some recent work showing that this zero free region is optimal. This is joint work with Arindam Roy.