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arXiv · 2406.16667

On the approximation of the zeta function by Dirichlet polynomials

Juan Arias de Reyna
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Abstract

We prove that for $s=σ+it$ with $σ\ge0$ and $0<t\le x$, we have \[ζ(s)=\sum_{n\le x}n^{-s}+\frac{x^{1-s}}{(s-1)}+Θ\frac{29}{14} x^{-σ},\qquad \frac{29}{14}=2.07142\dots\] where $Θ$ is a complex number with $|Θ|\le1$. This improves Theorem 4.11 of Titchmarsh.

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Juan Arias de Reyna. 2024-06-24. On the approximation of the zeta function by Dirichlet polynomials. https://arxiv.org/abs/2406.16667

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