The Superconducting Side: Odd Zeta Values and the Pole at s = 1 in BCS Theory

Summary: The weak-coupling theory of superconductivity of Bardeen, Cooper and Schrieffer (BCS) has a small set of universal numbers. These are the transition temperature T_c = (2e^γ/π) ω e^(−1/λ), the gap ratio 2Δ₀/T_c = 2π e^(−γ) = 3.528, the jump of the specific heat ΔC/C_n = 12/(7ζ(3)) = 1.426, and the two Werthamer–Helfand–Hohenberg (WHH) constants π² e^(−γ)/8 and 7ζ(3) e^(2−γ)/48 of the upper critical field. All of them have been known for decades. We show that they come from one function and two of its features. The function is (2^s − 1)ζ(s) = Σ_{n≥0} (n + ½)^(−s), the Dirichlet series of the half-integer lattice, on which the fermionic Matsubara frequencies (2n+1)πT live. The first feature is its finite part at the pole s = 1, which equals −ψ(½) = γ + 2 log 2. It fixes T_c for every common regularisation, and in the measurable ratio Δ₀/T_c the Euler factor 2 log 2 cancels, leaving π e^(−γ). The second feature is its values at odd integers. The gap equation and the clean gradient term are sums of odd powers |ω_n|^(−(2k+1)), so the Ginzburg–Landau coefficients contain exactly the odd values (1 − 2^(−2k−1))ζ(2k+1), and the 7 in 7ζ(3) is 2³ − 1. The radius of convergence of the Ginzburg–Landau series is the lowest Matsubara frequency πT. Pair breaking, by contrast, shifts the lattice and brings in all values ψ^(k)(½); the dirty WHH constant is ψ′(½) e^(ψ(1/2)) and contains the even value 3ζ(2). We record the stability of these statements under gap anisotropy (for d-wave pairing, Δ₀/T_c = 2π e^(−γ−1/2) and ΔC/C_n = 8/(7ζ(3))) and compare with Al, Nb and Pb. The tools are classical, and no new physics is claimed. What we add is the arithmetic order of known results, read side by side with the Fermi gas and the classical paramagnet of earlier papers. The paper is written for number theorists; Section 2 explains the physics from the beginning. The companion code (57 checks, Python with sympy and mpmath) reproduces every displayed constant and every numerical statement.