Exponentially damped reciprocal-prime sum
= Exponentially damped reciprocal-prime sum
For $3\leq z\leq D$, <Prime number theorem>[prime-number estimates] and <partial summation> give, for an absolute $c>0$,
$$
\sum_{p\leq z}\frac{e^{-\frac12\log D/\log p}}{p(\log p)^3}
\ll\frac1{(\log D)^3}e^{-c\log D/\log z}.
$$
Indeed, comparison with the prime-density integral and the substitution $v=\log D/\log t$ reduce the left side to $(\log D)^{-3}\int_{\log D/\log z}^\infty v^2e^{-v/2}\,dv$.