Zeros of partial sums of L-functions

We consider a certain class of multiplicative functions f:N→C. Let F(s)=∑ <inf>n=1</inf> ^∞ f(n)n ^−s be the associated Dirichlet series and F <inf>N</inf> (s)=∑ <inf>n≤N</inf> f(n)n ^−s be the truncated Dirichlet series. In this setting, we obtain new Halász-type results for the logarithmic mean value of f. More precisely, we prove estimates for the sum ∑ <inf>n=1</inf> ^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 the partial sums F <inf>N</inf> (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)=σ where σ≥1/2.