= Logarithmically weighted reciprocal-prime tail
{title2=$\sum_{p>x}\frac1{p(\log p)^\delta}\sim\frac1{\delta(\log x)^\delta}$}
For fixed $\delta>0$, the <Mertens first theorem> and <partial summation> imply convergence of $\sum_p1/(p(\log p)^\delta)$ 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 $1/\delta$.
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