Smoothed prime number theorem from a zero-free region

For a smooth compactly supported $F:(0,\infty )\to \mathbb{R}$ whose Mellin transform has suitable vertical decay, Mellin inversion andgiveMoving the contour through the pole at $s=1$ and into the classical zero-free region gives main term $X\tilde{F}(1)$ and error $O(X^{1-c/\log \log X})$ when $\tilde{F}(\sigma +it)\ll e^{-|t{|}^{1/2}}$.