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arXiv · 1308.0065

Zeros of partial sums of the Dedekind zeta function of a cyclotomic field

Andrew Ledoan·Arindam Roy·Alexandru Zaharescu
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Abstract

In this article, we study the zeros of the partial sums of the Dedekind zeta function of a cyclotomic field $K$ defined by the truncated Dirichlet series \[ ζ_{K, X} (s) = \sum_{\|\mathfrak{a}\| \leq X} \frac{1}{\|\mathfrak{a}\|^{s}}, \] where the sum is to be taken over nonzero integral ideals $\mathfrak{a}$ of $K$ and $\|\mathfrak{a}\|$ denotes the absolute norm of $\mathfrak{a}$. Specifically, we establish the zero-free regions for $ζ_{K, X} (s)$ and estimate the number of zeros of $ζ_{K, X} (s)$ up to height $T$.

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Andrew Ledoan, Arindam Roy, Alexandru Zaharescu. 2013-07-31. Zeros of partial sums of the Dedekind zeta function of a cyclotomic field. https://arxiv.org/abs/1308.0065

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