Moebius-convolutions and the Riemann hypothesis

The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and Hardy-Littlewood, based on the order of growth at infinity along the positive real axis of certain entire functions, are here imbedded in a general theorem for a class of entire functions, which in turn is seen to be a consequence of a rather transparent convolution criterion. Some properties of the convolutions involved sharpen what is hitherto known for the Riesz function.