Spectral properties of the Riemann zeta function and their physical applications

This work studies the Riemann $\zeta$ 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 ${\chi }^2\mathrm{/}\text{dof}=1.2$; (ii) the role of $\zeta (\frac{3}{2})$ 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 $\zeta$-function regularisation to the Casimir effect, yielding the standard Casimir energy density $-\frac{{\pi }^2\mathrm{\hslash }c}{720a^3}$, which for $a=1\mathrm{n}\mathrm{m}$ gives $\sim -4.33\times {10}^{-4}\mathrm{J}{\mathrm{m}}^{-2}$. 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.