How fast do the "partial sums of $\zeta(it)$" grow for fixed $t \neq 0$?

Question

In analytic number theory there are bounds for these partial sums, for example $$ \sum_{n \leq x} n^{-it} = O(x^{1/2} t^{\varepsilon}) $$ for $ |t| \geq x^{1/2} $ (so square root cancellation happens here as long as |t| is sufficiently large). But does it still hold when $t \neq 0$ is fixed and x goes to $\infty$? Is there something known about the growth of $ \sum_{n \leq x} n^{-it} $ for fixed $t \neq 0$ as $x \to \infty$?

Answer 1

The behavior as $x\to \infty$ is much easier, simply use $$\sum_{n\le x} n^{-it} = \sum_{n\le x} \int_n^{n+1}( x^{-it}+n^{-it}-x^{-it})dx=\sum_{n\le x} \int_n^{n+1}( x^{-it}+\int_n^x it y^{-it-1}dy)dx$$ $$= \sum_{n\le x} \int_n^{n+1}( x^{-it}+it x^{-it-1}+O(t^2 x^{-it-2}))dx = \frac{x^{1-it}}{1-it}+O(1+t^2)$$