  We consider the resolvent $(\lambda-a)^{-1}$ of any $R$-diagonal operator $a$
in a $\mathrm{II}_1$-factor. Our main theorem gives a universal asymptotic
formula for the norm of such a resolvent. En route to its proof, we calculate
the $R$-transform of the operator $|\lambda-c|^2$ where $c$ is Voiculescu's
circular operator, and give an asymptotic formula for the negative moments of
$|\lambda-a|^2$ for any $R$-diagonal $a$. We use a mixture of complex analytic
and combinatorial techniques, each giving finer information where the other can
give only coarse detail. In particular, we introduce {\em partition structure
diagrams}, a new combinatorial structure arising in free probability.
