Zeros of partial sums of $L$-functions

Abstract

We consider a certain class of multiplicative functions $f:\mathbb{N}\to \mathbb{C}$. 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 Halász-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 $\sigma >1/2$.

27 pages