OTHER REPRESENTATIONS OF THE RIEMANN ZETA FUNCTION AND AN ADDITIONAL REFORMULATION OF THE RIEMANN HYPOTHESIS

Abstract

New expansions for some functions related to the Zeta function in terms of the Pochhammer polynomials are given (coefficients $b_k$, $d_k$ and ${\widehat{d}}_k$). In some formal limit our expansion $b_k$ obtained via the alternating series gives the regularized expansion of Maslanka for the Zeta function. The real and the imaginary part of the function on the critical line is obtained with a good accuracy up to .

Then, we give the expansion (coefficient ${\widehat{d}}_k$) for the derivative of $\ln \left[\right(s-1\left)\zeta \right(s\left)\right]$. The critical function of the derivative, whose bounded values for at large values of $k$ should ensure the truth of the Riemann Hypothesis (RH), is obtained either by means of the primes or by means of the zeros (trivial and non-trivial) of the Zeta function. In a numerical experiment performed up to high values of $k$ i.e. up to $k={10}^{14}$ we obtain a very good agreement between the two functions, with the emergence of fourteen oscillations with stable amplitude.