On some analytic property of the Riemann zeta function

$$\int_x^\infty \{ u\} u^{-s-1} du = \int_x^\infty \left( \{ u\}-\frac{1}{2}\right) u^{-s-1} du + \frac{1}{2} \frac{x^{-s}}{s}$$ and by integration by parts,

$$\int_x^\infty \left( \{ u\}-\frac{1}{2}\right) u^{-s-1} du= - \int_x^\infty \left(\int_x^u \left( \{ t\}-\frac{1}{2}\right)dt \right) (-s-1) u^{-s-2} du=$$

$$= O\left( |s+1| \int_x^\infty u^{- \sigma -2 } \right) = O\left( |s+1| x^{-\sigma-1} / |\sigma +1| \right)$$

so the $\frac{1}{2} \frac{x^{-s}}{s}$ term dominates and you get a lower bound.