Eigenvalue Density, Li's Positivity, and the Critical Strip

TL;DR

This paper reformulates the zero-counting formula of the Riemann zeta function as a density distribution, deriving integral expressions for Li coefficients that relate to the zeros' criticality, without assuming the Riemann Hypothesis.

Contribution

It introduces a new formulation of the zero-counting formula as a density distribution and derives integral expressions for Li coefficients independent of the Riemann Hypothesis.

Findings

Li coefficients' positivity criterion is satisfied

Reformulation links zero distribution to physical interpretations

Proposes conjectures without assuming RH

Abstract

We rewrite the zero-counting formula within the critical strip of the Riemann zeta function as a cumulative density distribution; this subsequently allows us to formally derive an integral expression for the Li coefficients associated with the Riemann xi-function which, in particular, indicate that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We conjecture the validity of this and related expressions without the need for the Riemann Hypothesis and also offer a physical interpretation of the result and discuss the Hilbert-Polya approach.