Polylog / Riemann-zeta Bose–Einstein integrals: g_ν(z) = Li_ν(z); g_ν(1) = ζ(ν)

Bose–Einstein thermodynamic integrals as polylogarithms. Setup: ideal Bose gas in 3D with chemical potential μ < 0 and fugacity z = exp(μ/kT). Phase-space integrals ∫d³k/(exp(β(ε-μ))−1) reduce to Bose functions g_ν(z) = (1/Γ(ν))·∫₀^∞…