Non-analytic terms from nested divergences in maximal supergravity

The $D^4{R}^4$ and $D^6{R}^4$ coefficient functions in the effective action of type II string theory compactified on T ^d contain terms of the form $E_{1}ln{g}_d$ and $E_2{\left(ln{g}_d\right)}^2$ in specific dimensions, where g _d is the T-duality invariant string coupling, and $E_1$ and $E_2$ 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 $D^4{R}^4$ coupling, the contribution involves $E_{R^4}ln{g}_d$, while for the $D^6{R}^4$ coupling, it involves $E_{R^4}ln{g}_d$, $E_{D^2{R}^4}{\left(ln{g}_d\right)}^2$ and $E_{D^4{R}^4}ln{g}_d;$ where $E_{R^4}$, $E_{D^2{R}^4}$ and $E_{D^4{R}^4}$ are the $R^4$, $D^2{R}^4$ and $D^4{R}^4$ coefficient functions respectively. The contribution from $E_{D^2{R}^4}$, the coefficient function of an amplitude that vanishes onshell, arises from a two loop nested subdivergence of the three loop amplitude. (paper)