Riemann Hypothesis: The Riesz-Hardy-Littlewood wave in the long wavelength region

Abstract

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