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arXiv · 2009.12594

Real zeros of the Barnes double zeta function in the interval $(1, 2)$

Kazuma Sakurai
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Abstract

Let $a, w_1, w_2,\cdot\cdot\cdot, w_r >0$ and $s \in \mathbb{C}$. We put $w= (w_1,\cdot\cdot\cdot,w_r)$. Then the Barnes $r$-ple zeta function is defined by $ζ_r(s, w, a) = \sum_{m_1=0}^{\infty} \cdot\cdot\cdot \sum_{m_r=0}^{\infty} 1/(a+m_1w_1+\cdot\cdot\cdot +m_rw_r)^s$ when $σ:= \Re(s)>r$. In this paper, we show that the Barnes double zeta function $ζ_2(σ, w, a)$ has real zeros in the interval $(1,2)$ if and only if $0< a < (w_1+w_2)/2$ and the number of such zero is precisely one if $0< a< (w_1+w_2)/2$.

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Kazuma Sakurai. 2021-07-06. Real zeros of the Barnes double zeta function in the interval $(1, 2)$. https://arxiv.org/abs/2009.12594

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