Exact bounds on the closeness between the Student and standard normal distributions

Upper bounds on the Kolmogorov distance (and, equivalently in this case, on the total variation distance) between the Student distribution with p degrees of freedom (SD_p) and the standard normal distribution are obtained. These bounds are in a certain sense best possible, and the corresponding relative errors are small even for moderate values of p. The same bounds hold on the closeness between SD_p and SD_q with q>p.