Eigenvalue Density, Li's Positivity, and the Critical Strip
Keywords:Zero-counting formula, Riemann zeta function, Li coefficients
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.
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