We study the disordered Heisenberg spin chain, which exhibits many body localization at strong disorder, in the weak to moderate disorder regime. A continued fraction calculation of dynamical correlations is devised, using a variational extrapolation of recurrents. Good convergence for the infinite chain limit is shown. We find that the local spin correlations decay at long times as $C sim t^{-beta}$, while the conductivity exhibits a low frequency power law $sigma sim omega^{alpha}$. The exponents depict sub-diffusive behavior $ beta < 1/2, alpha> 0 $ at all finite disorders, and convergence to the scaling result, $alpha+2beta = 1$, at large disorders.
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