Violation of the Third Law of Thermodynamics by Black Holes, Riemann Zeta Function and Bose Gas in Negative Dimensions

Black holes violate the third law of thermodynamics in its standard formulation. Schwarzschild black hole entropy is inverse proportional to the square of the temperature $S=1\mathrm{/}(16\pi T^2)$ and tends to infinity rather than zero when the temperature goes to zero. We search for quantum statistical models with such exotic thermodynamic behaviour. It is shown that the Schwarzschild black hole in $D=4$ spacetime dimensions corresponds to a Bose gas in a space with $d=-4$ negative spatial dimensions. The Riemann zeta function is used to define the entropy of the Bose gas in negative dimension. The correspondence between black holes in higher dimensions and de Sitter spacetime with Bose gases is considered.