Zeroes of partial sums of the zeta-function

Abstract

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