Scattering amplitudes in string theory are written as formal integrals of correlations functions over the moduli space of punctured Riemann surfaces. It's well-known, albeit not often emphasized, that this prescription is only approximately correct because of the ambiguities in defining the integration domain. In this talk, we propose a resolution of this problem for one-loop open-string amplitudes and present their first evaluation at finite energy and scattering angle. Our method involves a deformation of the integration contour over the modular parameter to a fractal contour introduced by Rademacher in the context of analytic number theory. This procedure leads to explicit and practical formulas for the one-loop planar and non-planar type-I superstring four-point amplitudes, amenable to numerical evaluation. We plot the amplitudes as a function of the Mandelstam invariants and directly verify long-standing conjectures about their behavior at high energies.

Scattering amplitudes in string theory are written as formal integrals of correlations functions over the moduli space of punctured Riemann surfaces. It's well-known, albeit not often emphasized, that this prescription is only approximately correct because of the ambiguities in defining the integration domain. In this talk, we propose a resolution of this problem for one-loop open-string amplitudes and present their first evaluation at finite energy and scattering angle. Our method involves a deformation of the integration contour over the modular parameter to a fractal contour introduced by Rademacher in the context of analytic number theory. This procedure leads to explicit and practical formulas for the one-loop planar and non-planar type-I superstring four-point amplitudes, amenable to numerical evaluation. We plot the amplitudes as a function of the Mandelstam invariants and directly verify long-standing conjectures about their behavior at high energies.