New Formulas for the Euler-Mascheroni Constant and other Consequences derived from the Acceptance of Hyperbolicity of Jensen Polynomials and the Analysis of the Tur\'an Moments for the {\xi}-Function

TL;DR

This paper introduces new formulas for the Euler-Mascheroni constant derived from hyperbolicity of Jensen polynomials and Turán moments, supporting the Riemann Hypothesis and linking Gregory coefficients with the zeta function.

Contribution

It presents novel representations of the Euler-Mascheroni constant using Turán moments and Jensen polynomial coefficients, supporting hyperbolicity and the Riemann Hypothesis.

Findings

New formulas accurately approximate the Euler-Mascheroni constant.

Support for hyperbolicity of Jensen polynomials in the context of the Riemann Hypothesis.

Connections established between Gregory coefficients and the Riemann zeta function.

Abstract

The Euler-Mascheroni constant is calculated by three novel representations over these sets respectively: 1) Tur\'an moments, 2) coefficients of Jensen polynomials for the Taylor series of the Riemann Xi function at s=1/2+i.t and 3) even coefficients of the Riemann Xi function around s=1/2. These findings support the acceptance of the property of hyperbolicity of Jensen polynomials within the scope of the Riemann Hypothesis due to exactness on the approximations calculated not only for the Euler-Mascheroni constant but also for the Bernoulli numbers and the even derivatives of the Riemann Xi function at s=1/2. The new formulas are linked to similar patterns observed in the formulation of the Akiyama-Tanigawa algorithm based on the Gregory coefficients of second order and lead to understanding the Riemann zeta function as a bridge between the Gregory coefficients and other relevant sets