Quasi-Chaotic Property of the Prime-Number Sequence

The prime-number sequence, viewed as thespectrum of eigenvalues of random matrices, is found tobe quasi-chaotic. Plots of histograms of prime-numbernearest-neighbor spacing Delta p at various values of total number of integers indicate roughagreement with the Wigner distribution and illustratelevel repulsion. A global maximum of these curves isnoted at Δp = 6. Numerical work further implies that in any maximum integer sampling, no matterhow large, a finite number of nearest neighbor spacingsdo not occur. This quasichaotic property of theprime-number sequence supports the conjecture that a formula for the n-th prime does not exist. Arule for missing spacings is inferred according towhich, as maximum number of integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{N}$$ \end{document}increases, nearest neighbor vacancies corresponding tosmaller \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{N}$$ \end{document} vanish and new, larger value vacancies appear. Inaddition, early values of these histograms illustrate arough oscillatory behavior with periodicityδ[Δp] ≃ 6. A corollary to the resultsimplies that zeros of the Riemann zeta function likewise comprisea quasi-chaotic sequence. Application of these findingsto the resonant spectra of excited nuclei isnoted.