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

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

2009-03-25

arxiv.org/abs/0903.4321v2

Physics

Mathematical Physics

LaTeX, 18 pages; subtleties in convergence and branch cut added

Scientific paper

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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