Real zeros of Hurwitz zeta-functions and their asymptotic behavior in the interval $(0,1)$

Abstract

Let $0<a\le 1,s\in \mathbb{C}$, and $\zeta (s,a)$ be the Hurwitz zeta-function. Recently, T.~Nakamura showed that $\zeta (\sigma ,a)$ does not vanish for any $0<\sigma <1$ if and only if $1/2\le a\le 1$. In this paper, we show that $\zeta (\sigma ,a)$ has precisely one zero in the interval $(0,1)$ if $0<a<1/2$. Moreover, we reveal the asymptotic behavior of this unique zero with respect to $a$.

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