One Channel of Heat: ζ(2), Pendry's Bound and the Euler Factor at 2 under Temperature Doubling

Summary: When the temperature of black-body radiation doubles, its energy density grows by the factor 2^(d+1), which is 16 in three space dimensions. Writing the Bose occupation number as a sum over windings, 1/(e^x − 1) = Σ_{n≥1} e^(−nx), makes the origin of this factor transparent. The radiator at temperature T/2 consists exactly of the even windings of the radiator at T. Hence u(T)/(u(T) − u(T/2)) = (1 − 2^(−(d+1)))^(−1) is the Euler factor of ζ(d+1) at the prime 2, and inclusion–exclusion over the temperatures T/m produces any finite Euler product. For a single ballistic channel (d = 1) the heat current is ζ(2) k_B²T²/h, and the thermal conductance quantum is g₀ = 2ζ(2) k_B²T/h. It is known that g₀ is the same for phonons, photons and electrons. We show that the Bose–Fermi part of this universality is the identity 2η(2) = ζ(2): particles and holes of a degenerate Fermi channel together carry exactly the Bose value. This is the case s = 2 of 2η(s) − ζ(s) = (1 − 2^(2−s))ζ(s), and among real s > 1 this difference vanishes only at s = 2. Building on our earlier paper on the Fermi side, the same identity explains why the leading Sommerfeld coefficient is the only one that equals a zeta value, and why the Lorenz number is 2ζ(2)(k_B/e)². We prove a short lemma on Mellin moments of the entropy of a mode. It shows that Pendry's bound on the entropy current is attained by the thermal channel, and that the factor 2 in Ṡ = 2P/T is the ratio s/(s − 1) at weight s = 2. This also accounts for the factor 4 between Pendry's cooling bound and the actual power of a channel. Across dimensions the entropy per particle of the gas is (d+1)ζ(d+1)/ζ(d); it tends to 0 as d → 1 because ζ has its pole at 1. All physical formulas are classical and cited. Labels separate proved, cited and interpretive statements. Section 2 explains the physics from the beginning. The companion code (four Python scripts, 59 checks with mpmath at 30 digits, and the script for Figure 1) is included as P11_companion_code.zip. - Euler factor - Riemann zeta function - Dirichlet eta function - zeta(2) - Mellin transform - Möbius inversion - Bose–Einstein statistics - Fermi–Dirac statistics - black-body radiation - Stefan–Boltzmann law in d dimensions - thermal conductance quantum - single-channel heat transport - Pendry bound - entropy flow - Sommerfeld expansion - Wiedemann–Franz law - Lorenz number - particle–hole symmetry - pole at s = 1