Möbius convolutions and the Riemann hypothesis

Abstract

The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and of Hardy and Littlewood, based on the order of certain entire functions on the positive real axis, are here embedded 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.