  We consider a certain class of multiplicative functions $f: \mathbb N
\rightarrow \mathbb C$. Let $F(s)= \sum_{n=1}^\infty f(n)n^{-s}$ be the
associated Dirichlet series and $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ be the
truncated Dirichlet series. In this setting, we obtain new Hal\'asz-type
results for the logarithmic mean value of $f$. More precisely, we prove
estimates for the sum $\sum_{n=1}^x f(n)/n$ in terms of the size of
$|F(1+1/\log x)|$ and show that these estimates are sharp. As a consequence of
our mean value estimates, we establish non-trivial zero-free regions for these
partial sums $F_N(s)$.
  In particular, we study the zero distribution of partial sums of the Dedekind
zeta function of a number field $K$. More precisely, we give some improved
results for the number of zeros up to height $T$ as well as new zero density
results for the number of zeros up to height $T$, lying to the right of $\Re(s)
=\sigma$, where $\sigma > 1/2$.
