Zeros of partial sums of $L$-functions

TL;DR

This paper investigates the zeros of partial sums of $L$-functions, providing new zero-free regions and zero density estimates, especially for Dedekind zeta functions, through advanced mean value theorems for multiplicative functions.

Contribution

It introduces sharp Halász-type mean value estimates for multiplicative functions and applies these to establish improved zero-free regions and zero density results for partial sums of $L$-functions.

Findings

Established new zero-free regions for partial sums of $L$-functions.

Provided improved bounds on the number of zeros up to height $T$.

Derived zero density estimates for zeros to the right of $ ext{Re}(s)=\sigma > 1/2$.

Abstract

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 $\sigma > 1/2$.