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

TL;DR

This paper investigates the real zeros of the Barnes double zeta function within the interval (1, 2), establishing conditions on parameters for their existence and uniqueness.

Contribution

It provides a precise characterization of when the Barnes double zeta function has real zeros in (1, 2), including the exact number of such zeros under specific conditions.

Findings

Real zeros exist in (1, 2) if and only if 0 < a < (w_1 + w_2)/2.

There is exactly one zero in (1, 2) when 0 < a < (w_1 + w_2)/2.

Zeros are absent outside these parameter conditions.

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 $\zeta_r(s, w, a) = \sum_{m_1=0}^{\infty} \cdot\cdot\cdot \sum_{m_r=0}^{\infty} 1/(a+m_1w_1+\cdot\cdot\cdot +m_rw_r)^s$ when $\sigma := \Re(s)>r$. In this paper, we show that the Barnes double zeta function $\zeta_2(\sigma, w, a)$ 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 $0< a< (w_1+w_2)/2$.