= 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 formula>[Mellin inversion] and
$$
-\frac{\zeta'(s)}{\zeta(s)}=\sum_{n\geq1}\frac{\Lambda(n)}{n^s}
$$
give
$$
\sum_n\Lambda(n)F(n/X)
=\frac1{2\pi i}\int_{(2)}-\frac{\zeta'(s)}{\zeta(s)}\widetilde F(s)X^s\,ds.
$$
Moving the contour through the pole at $s=1$ and into the classical zero-free region gives main term $X\widetilde F(1)$ and error $O(X^{1-c/\log\log X})$ when $\widetilde F(\sigma+it)\ll e^{-|t|^{1/2}}$.
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