International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 21, Pages 3527-3537

A sequential Riesz-like criterion for the Riemann hypothesis

Luis Báez-Duarte

Departamento de Matemáticas, Instituto Venezolano de Investigaciones Científicas, Apartado Postal 21827, Caracas 1020-A, Venezuela

Received 7 June 2005; Revised 18 August 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.


Let ck:=j0k(1)j(kj)(1/ζ(2j+2)). We prove that the Riemann hypothesis is equivalent to ckk3/4+ε for all ε>0; furthermore, we prove that ckk3/4 implies that the zeros of ζ(s) are simple. This is closely related to M. Riesz's criterion which states that the Riemann hypothesis is equivalent to k=1((1)k+1xk/(k1)!ζ(2k))x1/4+ε as x+, for all ε>0.