Zeros of partial sums of L-functions

Abstract

We consider a certain class of multiplicative functions f:N→C. Let F(s)=∑∞n=1f(n)n−s be the associated Dirichlet series and FN(s)=∑n≤Nf(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 ∑xn=1f(n)/n in terms of the size of |F(1+1/logx)| 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 FN(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 R(s)=σ, where σ>1/2.