Journal Article

PRIMITIVE RIEMANN WAVE AT RE(S) = 0.9 AND APPLICATION OF THE GAUSS-LUCAS THEOREM

Sala, Massimo, Sala, Nicoletta, Merlini, Danilo

2020

Overview

Starting with a rigorous integral Equality, i.e., an integral over the logarithm of the modulus of the zeta function (an Equivalent of the Riemann Zeta Function having no zeros for Re(s) > 0.9) with a density (reciprocal of an hyperbolic function of parameter a, at Re(s) = 0.9), we define the Primitive Riemann Wave and compute some of its zeros (nodes). Alternatively, we study such a sequence of zeros by means of the Gauss -Lucas Theorem: attractive as well as repulsive points of the associated map are given up to t = 50, where t is the height in s = 0.9+i.t. We also report some numerical results for the case of vanishing parameter a, i.e., a→0 in the reciprocal hyperbolic measure. The numerical results suggest that the Primitive Riemann Wave has zeros extending to infinity i.e., that its convex hull contain in fact, the infinity of zeros of the Integrand, i.e., that no symmetry breaking occurs, supporting that the node of the Primitive Riemann Wave extends to infinity (with a vanishing amplitude if a > 0).

• Publisher: Nova Science Publishers, Inc
• Subject: Broken symmetry / Computational geometry / Convex hulls / Equality / Hyperbolic functions / Hypotheses / Infinity / Integrals / Parameters / Riemann waves / Symmetry / Theorems