The paper is D. J. Platt and T. S. Trudgian’s Zeroes of partial sums of the zeta-function, whose abstract reads:

This article considers the positive integers $N$ for which $\zeta_N(s)=\sum_{n=1}^Nn^{-s}$ has zeroes in the half-plane $\Re(s)>1$. Building on earlier results, we show that there are no zeroes for $1\le N\le18$ and for $N=20,21,28$. For all other $N$ there are infinitely many such zeroes.