Zeroes of partial sums of the zeta-function

David J. Platt; Timothy S. Trudgian

doi:10.1112/s1461157015000340

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arXiv · 1507.01340

Zeroes of partial sums of the zeta-function

David J. Platt·Timothy S. Trudgian

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Abstract

This article considers the positive integers $N$ for which $ζ_{N}(s) = \sum_{n=1}^{N} n^{-s}$ 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.

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David J. Platt, Timothy S. Trudgian. 2015-12-15. Zeroes of partial sums of the zeta-function. https://doi.org/10.1112/s1461157015000340

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