Logarithmically weighted reciprocal-prime tail (${\sum }_{p>x}\frac{1}{p(\log p{)}^{\delta }}\sim \frac{1}{\delta (\log x{)}^{\delta }}$)

For fixed $\delta >0$, the Mertens first theorem and partial summation imply convergence of ${\sum }_{p}1/(p(\log p{)}^{\delta })$ and, more precisely, the displayed tail asymptotic with error $O((\log x{)}^{-1-\delta })$. Write $A(t)={\sum }_{p\le t}(\log p)/p=\log t+O(1)$ and integrate against $(\log t{)}^{-1-\delta }$. The boundary at infinity vanishes; combining the finite boundary with the integral gives the coefficient $1/\delta$.