The Fermi Side: The Euler Factor at 2, the Sommerfeld Expansion and the Lorenz Number

Every constant of the ideal Fermi gas is built from the Dirichlet eta function η(s) = (1 − 2¹⁻ˢ)ζ(s). We make the arithmetic content of this fact explicit. In the Euler product of ζ(s) exactly one factor changes: the factor of the prime 2 becomes (1 − 2¹⁻ˢ)/(1 − 2⁻ˢ), all odd primes stay as they are. The prime is 2 because the Pauli principle cuts the occupation of a level after one particle, and a cut after q − 1 particles changes the factor of q. The new factor vanishes at s = 1 and removes the pole of ζ; η(1) = log 2. The moments of the derivative of the Fermi function are 2(2k)! η(2k), so the Sommerfeld coefficients are 2η(2k) = (2²ᵏ − 2)|B₂ₖ|π²ᵏ/(2k)!, and every coefficient of the low-temperature expansions of the heat capacity, the Lorenz number and the thermopower is a rational multiple of a power of π: the degenerate Fermi gas sees no odd zeta value. For integer order the Sommerfeld series terminates, and the remainder is exactly (−1)ⁿFₙ(−μ), an elementary form of the inversion formula for polylogarithms. We give the correct kinetic derivation of the Lorenz number L₀ = (π²/3)(k_B/e)² = 2.443004509… × 10⁻⁸ V² K⁻², the first temperature corrections for a power-law transport function (checked against a recent closed form), and the comparison with measured values. We separate three different factors of two: the Euler factor of the prime 2, the spin degeneracy, which cancels in L₀, and Drude’s arithmetic error of 1900. The same prime appears in the Fermi-to-Bose ratios 7/8 and 3/4 and in Boyer’s repulsive Casimir energy −7/8; we show that the Boyer factor ζ(s, ½)/ζ(s) = 2ˢ − 1 is minus the Fermi factor at 1 − s. Nothing in this paper is new physics, and most formulas are classical; the contribution is the bookkeeping and a list of statements that do not hold, among them that the electronic heat capacity tests the prefactor π²/3.