Zeros of partial sums of $L$-functions

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$\mathrm{.}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 ${\sum }_{n=1}^{x}f(n)\mathrm{/}n$ in terms of the size of $\mathrm{\mid }F(1+1\mathrm{/}\log x)\mathrm{\mid }$ 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$\sigma > 1/2$.