Exponentially damped reciprocal-prime sum

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 $(\log D{)}^{-3}{\int }_{\log D/\log z}^{\infty }{v}^2{e}^{-v/2}dv$.