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
Jacob's ladders, logarithmic modification of the Hardy-Littlewood integral (1918), Titchmarsh's $Ω$-theorem (1928) and new point of contact with the Fermat-Wiles theorem
Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918) and their relation to the Titchmarsh's sums (1943) and the Fermat-Wiles theorem
Jacob's ladders, Hardy-Littlewood integral (1918) and new asymptotic functional equations for Euler's Gamma function together with the tenth equivalent of the Fermat-Wiles theorem
Jacob's ladders, almost exact decomposition of certain increments of the Hardy-Littlewood integral (1918) by means of the Raabe's integral and the thirteenth equivalent of the Fermat-Wiles theorem
Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics
我们建立了$q(x_1,\cdots,x_n)=m$形式的仿射二次型中Zebriki开子集的Hardy-Littlewood性质（à la Borovoi-Rudnick），其中$q$是$n\geqslant 3$变量的非退化整二次型，$m$是非零整数.给出了取互质多项式值的整点密度的渐近公式，这是仿射二次曲面在无穷远处强逼近性质的算术纯性的一个定量形式。
Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918) and their relations to the Selberg's formula (1946) and the Fermat-Wiles theorem
