$T(\theta)=c_\beta^{-1}\ell(1/\theta)^{-1}\theta^{-\beta}(1+o(1))P+O(1)$.

\(a) This is part of [@AaronsonDenker01 Theorem 5.1]. (The main ideas of the proof are reproduced later in the proof of Lemma \[lem-ADerror\] and Lemma \[lem-zestimate\].)

\(b) By Proposition \[prop-PQ\], $$T(\theta)= (1-\lambda(\theta))^{-1}P(\theta)+
O(1)
=(1-\lambda(\theta))^{-1}P+
(1-\lambda(\theta))^{-1}(P(\theta)-P)+O(1).$$ By Corollary \[cor-H1\], $P(\theta)-P\ll \ell(1/\theta)\theta^\beta$, so the result follows from (a).

\(c) is immediate by (a) and (b).

The following expansion for $\lambda(\theta)$ will be used in proving results on second order asymptotics (Section \[sec-second\]).

\[lem-ADerror\] Suppose that $\mu(\varphi>n)=c(n^{-\beta}+H(n))$, where $c>0$ and $H(n)=O(n^{-q})$, $q>1$. Let $H_1(x)=[x]^{-\beta}-x^{-\beta}+H([x])$ and set $c_H=\int_0^\infty H_1(x)\,dx$. Then $$\lambda(\theta)=1-cc_\beta\theta^\beta+icc_H\theta+O(\theta^{2\beta},\theta^q),
\enspace\text{as $\theta\to0^+$}.$$

We follow [@AaronsonDenker01 Theorem 5.1]. Recall that $v(\theta)$ is the eigenfunction corresponding to $\lambda(\theta)$ normalised so that $\int v(\theta)\,d\mu=1$. Since $v(0)\equiv1$, it follows from Corollary \[cor-H1\] that $|v(\theta)-1|_\infty\ll \theta^\beta$. Let $\mathcal{F}_0$ denote the $\sigma$-algebra generated by $\varphi$. Define the step function $\hat v(\theta):[0,\infty)\to{{\mathbb C}}$ given by $\hat v(\theta)\circ \varphi=E(v(\theta)|\mathcal{F}_0)$ and the distribution function $G(x)=\mu(\varphi\le x)$. Then $$\begin{aligned}
\lambda(\theta) & =
{{\textstyle}\int}_Y R(\theta)v(\theta)\,d\mu={{\textstyle}\int}_Y e^{i\theta\varphi}v(\theta) \,d\mu 
=1+{{\textstyle}\int}_Y (e^{i\theta\varphi}-1)v(\theta)\,d\mu \\[.75ex] & =
1+{{\textstyle}\int}_0^\infty (e^{i\theta x}-1)\hat v_\theta(x)\,dG(x),\end{aligned}$$ where $\sup_{x\ge0}|\hat v_\theta(x)-1|\ll \theta^\beta$ and $1-G(x)=c(x^{-\beta}+H_1(x)).
$ Here $H_1(x)=O(x^{-q'})$ as $x\to\infty$, where $q'=\min\{q,\beta+1\}>1$. In particular, $H_1$ is integrable.

Write $\hat v_\theta=1+v_\theta^1-v_\theta^2+iv_\theta^3-iv_\theta^4$, where $v_\theta^s\ge0$ and $\sup_x |v_\theta^s(x)|\ll \theta^\beta$ for $s=1,2,3,4$. Define the positive measure $dG_\theta^s=v_\theta^s\, dG$. Then $$\begin{aligned}
