Albanian Journal of Mathematics (ISNN: 1930-1235), Vol 2, No 2 (2008)

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Riemann Hypothesis: A numerical treatment of the Riesz and hardy-Littlewood wave.

S. Beltraminelli, D. Merlini


We present the results of numerical experiments in connection with the Riesz and the Hardy-Littlewood criteria for the truth of the Riemann Hypothesis (RH). The coefficients $c_{k}$ of the Pochhammer expansion for the reciprocal of the Riemann Zeta function depend in our model on two parameters. The ``critical
functions'' $c_{k}k^{a}$ (where {\itshape a} is some constant), whose behaviour is concerned with the possible truth of the RH, are analysed at relatively large values of {\itshape k}. Some cases are numerically investigated up to larger values of {\itshape k}, i.e. $k={10}^{9}$ and more.

The $c_{k}$ we obtain in such a region have an oscillatory behaviour, which we call the Riesz and the Hardy-Littlewood wave. A special case is then studied numerically in some range of the critical strip. The numerical results give some evidence that the critical function is bounded for $\mathfrak{R}( s)
>\frac{1}{2}$ and such an ``evidence'' is stronger in the region $\mathfrak{R}( s) >\frac{3}{4}$ 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 ($\mathfrak{R}( s) >\frac{3}{4}$) and a (weaker) support in the direction to believe that the RH may be true ($\mathfrak{R}( s) >\frac{1}{2}$).

The amplitudes and the wavelength of the wave obtained by our numerical treatment are then compared with those formulated by Baez-Duarte in his analytical approach. The agreement is satisfactory.

Finally for another special case we found that the wave appears to be bounded even though one parameter in our model grows to infinity. Our analysis suggests that RH may barely be true and it is argued that an absolute bound on the amplitudes of the waves in all cases, should be given by $|\frac{1}{\zeta (
\frac{1}{2}+\epsilon ) }-1|$, with $\epsilon$ arbitrarily small positive, i.e. equal to 1.68477\ldots .

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