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

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 seriesζK,X(s)=∑‖a‖⩽X1‖a‖s, where the sum is to be taken over nonzero integral ideals a of K and ‖a‖ denotes the absolute norm of a. Specifically, we establish the zero-free regions for ζK,X(s) and estimate the number of zeros of ζK,X(s) up to height T.