Expand each logarithm as \log((1+T)\varepsilon_N^{\,c}-1)=\log_p(\varepsilon_N^{\,c}-1)+\log\!\big(1+\tfrac{\varepsilon_N^{\,c}T}{\varepsilon_N^{\,c}-1}\big), whose T^n-coefficient is \tfrac{(-1)^{n-1}}{n}\,\varepsilon_N^{\,cn}(\varepsilon_N^{\,c}-1)^{-n}. If (N,p)=1 then \varepsilon_N^{\,c}-1 is a p-adic unit, so this coefficient has valuation \ge -v_p(n); the coefficients grow only logarithmically and hence \widetilde F_\theta\in\sR^+. For general N=Dp^n, write \theta=\eta\chi and note, exactly as for F_\theta, the twisting formula \widetilde F_\theta(T)=-G(\chi^{-1})^{-1}\sum_{c\in(\Z/p^n\Z)^\times}\chi^{-1}(c)\,\widetilde F_\eta\big((1+T)\varepsilon_{p^n}^{\,c}-1\big). Since \widetilde F_\eta\in\sR^+ by the prime-to-p case and \sR^+ is stable under the substitution T\mapsto(1+T)\zeta-1 for a root of unity \zeta, the same holds for \widetilde F_\theta. This uses Definition 3.4.3.

This node stays unwired: the distribution-free route (replan R6.6) never needs full \sR^+-membership, only the weaker coefficient bound \|\,[T^n]\widetilde F_\theta\|\le C(n+1) that makes the boundary evaluations \widetilde F_\theta(\xi^i-1) converge (the Lean lemma summable_seriesEval_Ftilde, guarded by the norm-one fact for the contributing coprime roots).

Multiplication by x corresponds to the operator \partial=(1+T)\tfrac{d}{dT} on Mahler transforms, so x\,\widetilde\mu_\theta=\mu_\theta amounts to \partial\widetilde F_\theta=\sA_{\mu_\theta}=F_\theta. Differentiating termwise, \partial\log((1+T)\varepsilon_N^{\,c}-1)=\tfrac{(1+T)\varepsilon_N^{\,c}}{(1+T)\varepsilon_N^{\,c}-1}=1+\tfrac{1}{(1+T)\varepsilon_N^{\,c}-1}; summing against -G(\theta^{-1})^{-1}\theta^{-1}(c) the constant terms 1 cancel because \sum_c\theta^{-1}(c)=0 (as \theta is non-trivial), leaving exactly F_\theta(T). For the second statement, on \Zpx multiplication by x is invertible, and x\,\Res_{\Zpx}(\widetilde\mu_\theta)=\Res_{\Zpx}(\mu_\theta) gives \Res_{\Zpx}(\widetilde\mu_\theta)=x^{-1}\Res_{\Zpx}(\mu_\theta). This uses Lemma 6.3.1 and the Definition 3.4.3 dictionary under which multiplication by x corresponds to \partial.

We evaluate \big((1-\varphi\circ\psi)\widetilde F_\theta\big)(0) by computing the operator \varphi\circ\psi in two cases, using Lemma 6.3.2.

Case n>1 (so \chi\neq1). Since \chi vanishes on p\Zp, the twisted distribution \widetilde\mu_\theta=(\widetilde\mu_\eta)_\chi is supported on \Zpx, so \Res_{p\Zp}(\widetilde\mu_\theta)=0 and \varphi\circ\psi(\widetilde F_\theta)=0. Hence L_p(\theta,1)=\widetilde F_\theta(0), which we record as (1-\theta(p)p^{-1})\widetilde F_\theta(0) using \theta(p)=0.

Case n=0 (so N=D is coprime to p and \theta=\eta). Then \varphi\circ\psi(\widetilde F_\theta)(T)=\tfrac1p\sum_{\xi\in\mu_p}\widetilde F_\theta((1+T)\xi-1). Evaluating at T=0, the inner sum over \xi\in\mu_p collapses by the product identity \prod_{\xi\in\mu_p}(\xi\,\varepsilon_N^{\,c}-1)=\varepsilon_N^{\,pc}-1 (the roots of X^p-1; equivalently the norm of \xi\varepsilon_N^{\,c}-1 from \Q_p(\mu_p) to \Q_p), so \sum_{\xi}\log_p(\xi\varepsilon_N^{\,c}-1) =\log_p(\varepsilon_N^{\,pc}-1). Substituting the automorphism c\mapsto c'=pc of (\Z/N\Z)^\times (valid as p\nmid N) pulls out a factor \theta(p), giving \varphi\circ\psi(\widetilde F_\theta)(0)=\tfrac{\theta(p)}{p}\widetilde F_\theta(0). Therefore L_p(\theta,1)=(1-\theta(p)p^{-1})\widetilde F_\theta(0) again.

Finally, in all cases evaluate \widetilde F_\theta(0). Using \log_p(x)=\log_p(-x) for x\in\Cp^\times, we get \widetilde F_\theta(0)=-G(\theta^{-1})^{-1}\sum_c\theta^{-1}(c)\log_p(1-\varepsilon_N^{\,c}), and so L_p(\theta,1)=-(1-\theta(p)p^{-1})G(\theta^{-1})^{-1}\sum_c\theta^{-1}(c)\log_p(1-\varepsilon_N^{\,c}), as claimed.