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

2009/03/25 by Yang-Hui He, He, Yang-Hui, Vishnu Jejjala +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Number Theory (math.NT) #hep-th #math-ph #math.MP #math.NT

paper · pdf · doi:10.48550/arxiv.0903.4321

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

arxiv created 2009/04/22 · arxiv updated 2009/12/01

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.

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