Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 117 / 1 / d

Write $L=\log D$. By the Prime number theorem and partial summation, the sum is at most a constant timesMake the substitution $v=L/\log t$. Since $dt/t=-L{v}^{-2}dv$, this becomesFor $u>0$, the final integral is $O(e^{-cu})$ after decreasing the absolute constant $c>0$; any polynomial factor in $u$ is absorbed by the exponential, and bounded $u$ causes no problem. Taking $u=L/\log z$ provesthe exponentially damped reciprocal-prime sum estimate.