we verify that $\widehat T_n=T_n$ for all $n\ge0$. Throughout, $\beta\in(0,1)$.

\[lem-zestimate\] There exists $\epsilon,C>0$ such that $|1-\lambda(z)|\ge C\ell(1/\theta)\theta^\beta$ for all $z=\rho e^{i\theta}\in\bar{{\mathbb D}}\cap B_\epsilon(1)$.

We start off by mimicking the proof of Lemma \[lem-ADerror\]. Consider functions $v_z:[0,\infty)\to[0,\infty)$ satisfying either (i) $v_z\equiv1$ or (ii) $|v_z|_\infty=o(1)$ as $z\to1$. Write $z=e^{-u+i\theta}$, $0\le u,\theta<\epsilon$. Then the expansion $\lambda(z)-1$ leads to a linear combination of five integrals of the form $$\begin{aligned}
 \label{eq-five}
I=\int_0^\infty (e^{(-u+i\theta)x}-1)v_z(x)\,dG(x)=
(-u+i\theta)\int_0^\infty e^{(-u+i\theta)x}g_z(x)(1-G(x))dx,\end{aligned}$$ where $g_z\ge0$ and either (i) $g_z\equiv1$ or (ii) $|g_z|_\infty=o(1)$ as $z\to1$. Moreover there is one integral of type (i) and we show that this satisfies the desired lower bound, whilst the four integrals of type (ii) are negligible.

Recall that $1-G(x)=\mu(\varphi>x)=\ell([x])[x]^{-\beta}
=x^{-\beta}h(x)$ where $h(x)=\ell(x)(1+o(1))$. Substituting $\sigma=\theta x$, $$I=(-u+i\theta)\ell(1/\theta)
\theta^{\beta-1}\int_0^\infty e^{-\sigma y}e^{i\sigma}g_z(\sigma/\theta)
\sigma^{-\beta}h(\sigma/\theta)\ell(1/\theta)^{-1}\,d\sigma,$$ where $y=u/\theta$. We estimate the oscillatory integrals in the same way that alternating series are estimated in Leibnitz’s theorem, making extensive use of the fact that $\sigma\mapsto e^{-\sigma y}g_z(\sigma/\theta)
\sigma^{-\beta}h(\sigma/\theta)$ is decreasing for each fixed $\theta,y$.

Write $|I|= (u^2+\theta^2)^{1/2}\ell(1/\theta)\theta^{\beta-1}A$, where in cases (i) and (ii) respectively, $$\begin{aligned}
A_{(i)}& \ge \int_0^{3\pi/2} \cos\sigma\, e^{-\sigma y}\sigma^{-\beta}h(\sigma/\theta)\ell(1/\theta)^{-1}\,d\sigma, \\
A_{(ii)} & \le 
 \Bigl(\int_0^{\pi/2}\cos\sigma+\int_0^{\pi}\sin\sigma\Bigr)e^{-\sigma y}
\sigma^{-\beta}
g_z(\sigma/\theta)h(\sigma/\theta)\ell(1/\theta)^{-1}\,d\sigma.\end{aligned}$$

We divide the region $y>0$ into the regions $y\in(0,1/\delta]$ and $y\ge1/\delta$ where $\delta$ is chosen sufficiently 