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Logarithmic corrections in the asymptotic expansion for the radiation field along null infinity

We obtain the second-order asymptotics for the radiation field of spherically symmetric solutions to the wave equation on spherically symmetric and asymptotically flat backgrounds including the Schwarzschild and sub-extremal Reissner-Nordstrom families of black holes. These terms appear as logarithmic corrections to the leading-order asymptotic terms which were rigorously derived in our previous work. Such corrections were heuristically and numerically derived in the physics literature in the case of a non-vanishing Newman-Penrose constant. In this case, our results provide a rigorous confirmation of the existence of these corrections. On the other hand, the precise logarithmic corrections for compactly supported initial data (and hence with a vanishing Newman-Penrose constant) explicitly obtained here appear to be new.