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

TL;DR

This paper introduces new Pochhammer polynomial-based expansions related to the Riemann Zeta function, providing accurate approximations of its real and imaginary parts on the critical line and proposing a reformulation of the Riemann Hypothesis.

Contribution

It presents novel expansions of the Zeta function and its derivatives using Pochhammer polynomials, offering a new perspective and numerical evidence related to the Riemann Hypothesis.

Findings

Accurate approximation of Zeta function on the critical line up to Im(s) < 35

Numerical experiments up to k=10^13 show stable oscillations in critical functions

Proposes a bound on the critical function related to Euler's constant gamma

Abstract

New expansions for some functions related to the Zeta function in terms of the Pochhammer's polynomials are given (coefficients b(k), d(k), d_(k) and 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 Im(s) = t < 35. Then, we give the expansion (coefficient d_(k)) for the derivative of ln((s-1)\zeta(s)). The critical function of the derivative, whose bounded values for Re(s) > 1/2 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 = 10^13 we obtain a very good agreement between the two functions, with the emergence of twelve oscillations with stable amplitude. For a special case of values of the two parameters entering in the general Pochhammer's expansion it is argued that the bound on the critical function should be given by the Euler constant gamma.