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New expansions for some functions related to the Zeta function in terms ofthe Pochhammer's polynomials are given (coefficients b(k), d(k), d_(k) andd__(k). In some formal limit our expansion b(k) obtained via the alternatingseries gives the regularized expansion of Maslanka for the Zeta function. Thereal and the imaginary part of the function on the critical line is obtainedwith a good accuracy up to Im(s) = t < 35.  Then, we give the expansion (coefficient d_(k)) for the derivative ofln((s-1)\zeta(s)). The critical function of the derivative, whose boundedvalues for Re(s) > 1/2 at large values of k should ensure the truth of theRiemann Hypothesis (RH), is obtained either by means of the primes or by meansof the zeros (trivial and non-trivial) of the Zeta function. In a numericalexperiment performed up to high values of k i.e. up to k = 10^13 we obtain avery good agreement between the two functions, with the emergence of twelveoscillations with stable amplitude.  For a special case of values of the two parameters entering in the generalPochhammer's expansion it is argued that the bound on the critical functionshould be given by the Euler constant gamma.

Other representations of the Riemann Zeta function and an additional reformulation of the Riemann Hypothesis