The Classical Side: Zeta Values in Curie's Law, the Brillouin Function and Molecular Rotation

Summary: In quantum statistics the zeta values are in plain view: Planck's law carries ζ(4), the Fermi gas carries η(s). Classical statistics seems to have lost them. Curie's law for a paramagnet has the rational slope 1/3, and the high-temperature series of the rotational partition function of a diatomic molecule, T/Θ_R + 1/3 + (1/15)Θ_R/T + …, has rational coefficients as well. We show where the zeta values sit. First, the partition function of a classical magnetic moment is sinh x/x, an entire function with zeros at iπk, k ≠ 0. Its Weierstrass product makes every Taylor coefficient of the Langevin function a value ζ(2n)/π²ⁿ; Curie's 1/3 is 2ζ(2)/π². The same function is the ratio of the classical and the quantum partition function of one harmonic oscillator, and its zeros are bosonic Matsubara frequencies. Second, a quantum spin J keeps only the zeros iπk with m ∤ k, m = 2J+1, in a suitable variable. This removes the multiples of one modulus and gives the factor 1 − m⁻²ⁿ; for n = 1 it is the Curie factor J(J+1) = (m²−1)/4 of every textbook. Spin 1/2 keeps the odd zeros only, which is the factor 1 − 2⁻²ⁿ of the Fermi side. Among classical moments in d dimensions only d = 1 and d = 3 have their zeros on a translate of πℤ. Third, the rotational series comes from Hurwitz values ζ(−2k−1, 1/2). If one separates even and odd rotational quantum numbers, as nuclear spin does in para- and ortho-hydrogen, the difference is exactly e^(σ/4)·η(iσ/π)³ with the Dedekind eta function. Hence both parts have the same asymptotic series, and the deviation of the equilibrium ortho:para ratio from 3:1, 0.35% at room temperature, is a modular form. We show that the inversion σ ↦ π²/σ behind this, which exchanges high and low temperature around T* = Θ_R/π, is after a Mellin transform exactly the functional equation w ↔ 1−w of L(w, χ₋₄), while the full rotational sum has no such law. Geometrically, the rotational partition function is the heat trace of the sphere, para-hydrogen is the projective plane, and the exponent π²/(4σ) is the squared distance between antipodal points. The tools are classical: Mittag-Leffler, Zagier's shifted Euler–Maclaurin formula and Jacobi's identity for η³. The paper is written for number theorists; Section 2 explains the physics from the beginning. No new physics is claimed. The companion code (48 checks, Python with sympy and mpmath) reproduces every displayed coefficient and every numerical statement. Riemann zeta function - zeta values at even integers - Bernoulli numbers - Hurwitz zeta function - Dirichlet L-function - functional equation - Mellin transform - Dedekind eta function - theta series - modular forms - Weierstrass product - Mittag-Leffler expansion - heat trace - Curie's law - Langevin function - Brillouin function - paramagnetism - rotational partition function - ortho- and para-hydrogen - Matsubara frequencies