Normalized spacings between zeros of Riemann zeta function follow normalized Maxwell-Boltzmann distribution

ORAL

Abstract

Through \textit{Planck} relation $\varepsilon =$ h$\nu $ normalized spacings between energy levels of oscillators are related to those between frequencies expressed as \textit{Gauss} clock calculator or \textit{Hensel} p$_{\mathrm{j}}$-adic numbers. Energy-level spacings are related to spacings between ``stationary states'' and through \textit{Euler} golden key to zeros of \textit{Riemann} zeta function. The latter are shown to follow normalized \textit{Maxwell-Boltzmann} (NMB) distribution function, \begin{equation} \rho_{\beta} = (8/\pi_{\beta}) [(2/\sqrt{\pi_{\beta}} )x_{\beta} ]^{2} e^{-[(2/\sqrt{\pi_{\beta}} )x_{\beta}]^{2}} \end{equation} , hence providing physical explanations of \textit{Montgomery-Odlyzko} law and \textit{Hilbert-Polya} conjecture. Position of the critical line is found to coincide with that of stationary states. Normalized spacing between eigenvalues of GUE of an \textit{Adele} space constructed by superposition of infinite NMB distribution functions will coincide with spacing of zeros of \textit{Riemann} zeta function according to the theory of noncommutative geometry of \textit{Connes}.

Authors

  • Siavash Sohrab

    Northwestern University