At low energies, interactions of massless particles in type II strings compactified on a torus $T^d$ are described by an effective Wilsonian action $mathcal{S}(Lambda)$, consisting of the usual supergravity Lagrangian supplemented by an infinite series of higher-derivative vertices, including the much studied $ abla^{4p+6q} mathcal{R}^4$ gravitational interactions. Using recent results on the asymptotics of the integrands governing four-graviton scattering at genus one and two, I determine the $Lambda$-dependence of the coefficient of the above interaction, and show that the logarithmic terms appearing in the limit $Lambdato 0$ are related to UV divergences in supergravity amplitudes, augmented by stringy interactions. This provides a strong consistency check on the expansion of the integrand near the boundaries of moduli space, in particular it elucidates the appearance of odd zeta values in these expansions. I briefly discuss how these logarithms are reflected in non-analytic terms in the low energy expansion of the string scattering amplitude.
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