Non--analytic terms from nested divergences in maximal supergravity

The $D^4\mathcal{R}^4$ and $D^6\mathcal{R}^4$ coefficient functions in the effective action of type II string theory compactified on $T^d$ contain terms of the form $\mathcal{E}_1 {\rm ln}g_d$ and $\mathcal{E}_2 ({\rm ln}g_d)^2$ in specific dimensions, where $g_d$ is the T--duality invariant string coupling, and $\mathcal{E}_1$ and $\mathcal{E}_2$ are U--duality invariant coefficient functions. We derive these non--analytic terms from nested ultraviolet divergences in two and three loop maximal supergravity. For the $D^4\mathcal{R}^4$ coupling, the contribution involves $\mathcal{E}_{\mathcal{R}^4} {\rm ln} g_d$, while for the $D^6\mathcal{R}^4$ coupling, it involves $\mathcal{E}_{\mathcal{R}^4} {\rm ln} g_d$, $\mathcal{E}_{D^2\mathcal{R}^4} ({\rm ln} g_d)^2$ and $\mathcal{E}_{D^4\mathcal{R}^4} {\rm ln} g_d$; where $\mathcal{E}_{\mathcal{R}^4}$, $\mathcal{E}_{D^2\mathcal{R}^4}$ and $\mathcal{E}_{D^4\mathcal{R}^4}$ are the $\mathcal{R}^4$, $D^2\mathcal{R}^4$ and $D^4\mathcal{R}^4$ coefficient functions respectively. The contribution from $\mathcal{E}_{D^2\mathcal{R}^4}$, the coefficient function of an amplitude that vanishes onshell, arises from a two loop nested subdivergence of the three loop amplitude.