Zeroes of partial sums of the zeta-function

Abstract

This article considers the positive integers $N$ for which ${\zeta }_N(s)={\sum }_{n=1}^N{n}^{-s}$ has zeroes in the half-plane $\Re (s)>1$. Building on earlier results, we show that there are no zeroes for $1\le N\le 18$ and for $N=20,21,28$. For all other $N$ there are infinitely many zeroes.

arXiv comment5 Pages - Final Version will appear in LMS JCM