Want to show $\sum_{n>N}n^{-1/2} \ll N^{-1/2}$

$$\sum_{n=x}^N \frac{a_n}{n^s} = A(N)N^{-s} + s \int_x^N A(t)t^{-s-1}dt$$ gives for $\Re(s) >0$ $$|\sum_{n=x}^\infty \frac{\chi(n)}{n^s}| =| s \int_x^\infty (\sum_{x\leq n\leq t}\chi(n))t^{-s-1}dt|\le |s| \int_x^\infty q t^{-\Re(s)-1}dt= \frac{q|s| x^{-\Re(s)}}{\Re(s)}$$