  We present the results of numerical experiments in connection with the Riesz
and Hardy-Littlewood criteria for the truth of the Riemann Hypothesis (RH). The
coefficients c_k of the Pochammer's expansion for the reciprocal of the Riemann
Zeta function, as well as the ``critical functions'' c_k*k^a (where a is some
constant), are analyzed at relatively large values of k. It appears an
oscillatory behaviour (Riesz-Hardy-Littlewood wave). The amplitudes and the
wavelength of the wave are compared with an analytical treatment concerning the
wave in the asymptotic region. The agreement is satisfactory. We then find
numerically that in the large beta limit too, the amplitudes of the waves
appear to be bounded. For a special case the numerical experiments are
performed up to larger values of k, i.e k=10^9 and more. The analysis suggests
that RH may barely be true and an absolute bound for the amplitudes of the
waves in all cases should be given by |1/(zeta(1/2)+epsilon) -1|, with epsilon
arbitrarily small positive, i.e. equal to 1.68
