Bose–Einstein Condensation and the Debye Law The Density of States Chooses the Zeta Argument, the Pole Decides

For an ideal Bose gas whose density of states grows like εᵃ⁻¹, every equilibrium quantity at the condensation point is a Mellin moment Γ(a)ζ(a) or Γ(a+1)ζ(a+1). The exponent is a = d/2 + d/δ for a gas in d dimensions in a potential r^δ: a = 3/2 in a box, a = 3 in a harmonic trap. We make the arithmetic content of this classical rule explicit. The density of states chooses the zeta argument, and the only pole of ζ, at s = 1, decides three things. Condensation occurs if and only if a > 1, because Liₐ(1) = ζ(a) is finite. The heat capacity jumps at T_c if and only if a > 2, by a²ζ(a)/ζ(a−1), because ζ(a−1) is finite; the value below T_c is a(a+1)ζ(a+1)/ζ(a). In the borderline case a = 1, the one-dimensional harmonic trap, the pole turns into a logarithm: the excited-state sum is the Lambert series Σ d(n)e^(−n/t) of the divisor function with Mellin transform Γ(s)ζ(s)², and its double pole gives t(log t + γ) + 1/4 − 1/(144t) + …, with 1/4 = ζ(0)² and 1/144 = ζ(−1)². The finite-size shift −ζ(2)/(2ζ(3)^(2/3)) · N^(−1/3) of a three-dimensional trap is the next residue of the same kind of transform. Fermions never condense, because η(s) has no pole. The Debye law belongs to the same scheme with an even argument: 12π⁴/5 = 216 ζ(4), and its T³ form is exact at low temperature with the directional average of Σᵢ cᵢ⁻³ over the three sound branches. For copper at 0 K the elastic constants give Θ_D = 344.4 K against the calorimetric 343 K, an independent test of the prefactor at the level of one per cent. The value ζ(3/2) of the box has no known closed form. We also record that the first condensates of 1995 were made in harmonic traps and so test ζ(3), not ζ(3/2). Accompanying code repository contents (SHA256SUMS): a14986048687f94245c87d98670266179a8671cbeeae4af368a675388f778ede README.mdd685a7bcfe008d42d08d4f1546809ddfba3a5bcd4244a92b905b8f7e0ca76c6f p4_checks.py67eca726ed7311cedaf292189e4c55a6cc1c8ca54763ed2e91359617050a4660 p4_checks.log1be23f68b50ea329f2e74db39924ccc60902b3c2db149b921af8d2995fd304ca run_all_checks.py63de8f343011fd3c5e921f67371a96630a5b4ccfaf4194877fd10d963276da61 run_all_checks.log