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A density of states $g(ϵ)=g_0[1+\alpha \cos (2\pi ϵ/\Delta )]$ has zero-temperature oscillatory grand potential $g_0\alpha {\Delta }^2\cos (2\pi \mu /\Delta )/(4{\pi }^2)$. Its Fermi-window convolution multiplies that oscillation by $R_T$ when the lower band edge is far below the chemical potential. The Fourier transform of the even window removes its sine component and gives the title factor. At high $X$, $R_T\sim 2X{e}^{-X}$; at small $X$, $R_T=1-X^2/6+O(X^4)$. The corresponding thermal change relative to zero temperature has amplitude proportional to $1-R_T$, rather than $R_T$.

The formal product over fermionic Matsubara frequencies is divergent before its normalization and time-slice prescription are fixed. The coherent-state thermal trace fixes the single-level factor to $1+e^{-\beta \xi }$, counting its empty and occupied states. A symmetrically regularized product can instead expose $2\cosh (\beta \xi /2)$; the additional factor $e^{-\beta \xi /2}$ restores the physical empty-state energy convention. Dropping that factor can incorrectly shift the grand potential and particle occupation.