Zeros of partial sums of the Dedekind zeta function of a cyclotomic field

Abstract

In this article, we study the zeros of the partial sums of the Dedekind zeta function of a cyclotomic field $K$ defined by the truncated Dirichlet series \[ ζ_{K, X} (s) = \sum_{\|\mathfrak{a}\| \leq X} \frac{1}{\|\mathfrak{a}\|^{s}}, \] where the sum is to be taken over nonzero integral ideals $\mathfrak{a}$ of $K$ and $\Vert \mathfrak{a}\Vert$ denotes the absolute norm of $\mathfrak{a}$. Specifically, we establish the zero-free regions for ${\zeta }_{K,X}(s)$ and estimate the number of zeros of ${\zeta }_{K,X}(s)$ up to height $T$.

arXiv comment12 pages, Submitted for publication