Abel summation formula

Partial summation is the discrete analogue of integration by parts. If $A(x)={\sum }_{n\le x}{a}_n$ and $g$ is continuously differentiable, then

For $3\le z\le D$, prime-number estimates and partial summation give, for an absolute $c>0$,Indeed, comparison with the prime-density integral and the substitution $v=\log D/\log t$ reduce the left side to $(\log D{)}^{-3}{\int }_{\log D/\log z}^{\infty }{v}^2{e}^{-v/2}dv$.