  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.
