International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 22, Pages 3599-3608
Möbius convolutions and the Riemann hypothesis
Departamento de Matemáticas, Instituto Venezolano de Investigaciones
Científicas, Apartado Postal 21827, Caracas 1020-A, Venezuela
Received 7 June 2005; Revised 6 October 2005
Copyright © 2005 Luis Báez-Duarte. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.