Some software packages make use of the following definition for generalized harmonic numbers. In what follows, $\sigma,t\in\mathbb{R}$:
$$H_{ t }^{(\sigma+it)}=\zeta (\sigma+it)-\zeta \ (\sigma+it, t +1)$$
where $\zeta(x)$ is the Riemann zeta function and $\zeta(x,y)$ is the Hurwitz zeta function. Also, some of these packages use the following default evaluation:
$$\sum _ {m=1}^{\lfloor t \rfloor} \ \frac{1}{m^{\sigma+it}}=H_{\lfloor t \rfloor }^{(\sigma+it)}=\zeta (\sigma+it)-\zeta \ (\sigma+it, \lfloor t \rfloor +1)$$
My question is whether this last identity above is a known and proven identity; in other words, is the sum really exactly and provably equal to the rightmost expression in terms of the Riemann zeta and the Hurwitz zeta