Investigation about a statement equivalent to Riemann Hypothesis (RH)

TL;DR

This paper explores a novel approach to the Riemann Hypothesis by analyzing a quantity akin to angular momentum related to the Xi function, linking it to prime number distribution and zero locations.

Contribution

It introduces a new equivalence statement involving the Xi function's maxima and minima, and uses Euler products and prime number theorems to support the hypothesis.

Findings

Positivity condition holds for Xi on the critical line

Off-critical zeros are excluded under the proposed equivalence

Prime spectrum convergence is highlighted as a by-product

Abstract

First idea is to compute a quantity like the angular momentum with respect to (0, 0), of an unitary mass of coordinates (<[Xi(s)], =[Xi(s)]) while =[s] is the time, and, <[s] = constant. If we impose that the derivative along <[s], at points <[s] = 1/2 is grater than zero, then, we find exactly a known RH equivalence statement about relative maxima and minima of Xi(1/2 + i=[s]) along critical line. After representing this fictitious angular momentum by Euler Product, and, using PNT as a tool, it can be proved that this positivity condition is granted everywhere at least for Xi(1/2 + i=[s]) 6 = 0. So, if the above equivalence is true, it is found that off-critical line zeros must be excluded for Z(s) function along all critical strip . Further analysis on Euler Product(Lemma 2) has evidenced others shorter ways to same objective. Besides the converging spectrum of prime numbers is highlighted as a by-product.