Spectral properties of the Riemann zeta function and their physical applications

Abstract

This work studies the Riemann $ζ$ numerically with emphasis on its physical manifestations. Three areas are examined: (i) the spectral statistics of the non-trivial zeros, which we compare with the Gaussian Unitary Ensemble predictions, obtaining $χ^2/\text{dof} = 1.2$; (ii) the role of $ζ(\frac{3}{2})$ in determining the critical temperature for Bose-Einstein condensation, where we find $T_c = 0.13$ K for an ideal gas; and (iii) the application of $ζ$-function regularisation to the Casimir effect, yielding the standard Casimir energy density $-\frac{π^2 \hbar c}{720 a^3}$, which for $a=1\,\mathrm{nm}$ gives $\sim -4.33\times10^{-4}\,\mathrm{J\,m^{-2}}$. The numerical results are consistent with the Montgomery-Odlyzko conjecture and with spectral approaches motivated by the Hilbert-Pólya conjecture, within the statistical limitations of the analysis.