Zeroes of partial sums of the zeta-function

TL;DR

This paper investigates the zeros of partial sums of the Riemann zeta function, establishing the absence of zeros for small N and their abundance for larger N, revealing new insights into their distribution.

Contribution

It demonstrates the exact N values where the partial sums have no zeros and proves the existence of infinitely many zeros for all other N, extending previous results.

Findings

No zeros for 1 ≤ N ≤ 18, 20, 21, 28

Infinitely many zeros for all other N

Extends understanding of zeros 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 $1\leq N\leq 18$ and for $N=20, 21, 28$. For all other $N$ there are infinitely many zeroes.