Riemann Hypothesis: a special case of the Riesz and Hardy-Littlewood wave and a numerical treatment of the Baez-Duarte coefficients up to some billions in the k-variable

Abstract

We consider the Riesz and Hardy-Littlewood wave i.e. a ``critical function'' whose behaviour is concerned with the possible truth of the Riemann Hypothesis (RH). The function is studied numerically for the case alpha = 15/2 and beta = 4 in some range of the critical strip, using Maple 10. In the experiments, N = 2000 is the maximum argument used in the Moebius function appearing in c_k i.e. the coefficients of Baez-Duarte, in the representation of the inverse of the Zeta function by means of the Pochammer's polynomials. The numerical results give some evidence that the critical function is bounded for Re(s) > 1/2 and such an ``evidence'' is stronger in the region Re(s) > 3/4} where the wave seems to decay slowly. This give further support in favour of the absence of zeros of the Riemann Zeta function in some regions of the critical strip (Re(s) > 3/4) and a (weaker) support in the direction to believe that the RH may be true (Re(s) > 1/2).