It is known from, for example, Apostol (Introduction to Analytic Number Theory), Theorem 12.23, that there exists a constant $A$ not depending on $t$ and $a$ such that for $|t|\ge 1$, the Hurwitz zeta function $\zeta(-0.5+it,a)$ can be bounded by $$|\zeta(-0.5+it,a)|\le A|t|^{5/2}.$$
Are there any bounds for $|\zeta(-0.5+it,a)|$ with explicit constant $A$ and smaller power of $|t|$ whenever $|t|\ge 1$?
Also, I am wondering if there is a uniform bound for $|\zeta(-0.5+it,a)|$ whenever $|t|\le 1$ and $0<a\le 1$? (Settled)
I appreciate it a lot if someone could provide relevant references.
For $a \ge 1$ and $\Re(s) > 1$ and by analytic continuation for $\Re(s) > 0$ and $\Re(s) > -1$ $$\zeta(s,a) - \frac{a^{1-s}}{s-1}-\frac{s}2\zeta(s+1,a) =\sum_{n\ge 0} (n+a)^{-s}-\int_{n+a}^{n+a+1} x^{-s}dx-\frac{s}2(n+a)^{-s-1}$$ $$=\sum_{n\ge 0}\int_{n+a}^{n+a+1}\int_{n+a}^x (st^{-s-1}-s(n+a)^{-s-1})dtdx$$
$$=-\sum_{n\ge 0}\int_{n+a}^{n+a+1}\int_{n+a}^x s\int_{n+a}^t (s+1)u^{-s-2}dudtdx$$ ie.
$$|\zeta(s,a) - \frac{a^{1-s}}{s-1}-\frac{s}2\zeta(s+1,a)|\le |s(s+1)|\sum_{n\ge 0} (n+a)^{-\Re(s)-2}\le |s(s+1)| (1+\frac{1}{\Re(s)+1})$$
Since $\zeta(s,a)=\zeta(s,a+1)+a^{-s}$ it gives $$\forall t\in \Bbb{R},a > 0,\qquad |\zeta(-1/2+it,a)| \le C (1+t^2)$$