From arithmetic spectra to a quantum-corrected black hole geometry

Abstract

The Euler product of the Riemann zeta function is the partition function of a free bosonic gas whose mode energies are the logarithms of the primes. We show that this arithmetic gas, combined with the assumption that the entropy--geometry correspondence holds, leads to a quantum-corrected black-hole metric. The prime gas is a Hagedorn system. Its entropy is linear in the energy, which is exactly what an entropy linear in the horizon area requires, and the simple pole of the zeta function at $\beta=1$ fixes the coefficient of the logarithmic correction to the area law. The nontrivial zeros cannot play this role: their level density grows only logarithmically, and far too slowly to be extensive. Demanding that the reconstructed geometry reduce to Schwarzschild at large radius then fixes the map from arithmetic energy to horizon area and yields the closed-form metric $f(r)=1-2GMr/(r^{2}+\lz^{2})$ with $\lz^{2}=\alpha G/\pi$. This describes a two-horizon black hole with Reissner--Nordstr\"om horizon structure but no Coulombic hair, a positive-energy anisotropic source obeying the null energy condition, a softened central singularity, a bounded Hawking temperature, and a cold extremal remnant that ends the evaporation. The same length scale follows independently from requiring that the first law hold exactly with the corrected entropy. The nontrivial zeros survive only as exponentially suppressed log-periodic ripples in the area, which suggests a physical interpretation of the Riemann hypothesis as the statement that arithmetic corrections to black-hole thermodynamics are as small as they can be.