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

Let $0<a\leq1, s\in\mathbb{C}$, 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 $1/2\leq a \leq1$. 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 $a$.