not dominated by ${\operatorname{Re}}\{(1-\lambda(z))^{-1}\}$. As a consequence, ${\operatorname{Re}}T(z)={\operatorname{Re}}T(\rho e^{i\theta})$ is not dominated by a single integrable function of $\theta$, see Lemma \[lem-R1\] below.

We have $\mu(\varphi>n)=\ell(n)n^{-1}$, where $\ell$ is slowly varying and . Let $\tilde\ell(x)=m(x)=\sum_{j=1}^{[x]} \ell(j)j^{-1}$. Then $\tilde\ell$ is monotone increasing and $\lim_{n\to\infty}\tilde\ell(n)=\infty$. By Proposition \[prop-Karamata\](b), $\tilde\ell$ is slowly varying and $\ell(n)/\tilde\ell(n)\to0$ as $n\to\infty$. Up to asymptotic equivalence, we have the alternative definitions $\tilde\ell(x)=\int_1^x  \ell(y)y^{-1}\,dy$ and $\tilde\ell(x)=\int_0^x  (1-G(y))\,dy$ where $G(x)=\mu(\varphi\le x)$.

\[prop-elltilde\] ${\displaystyle}\int_0^{1/y} \frac{\ell(1/\theta)}{\theta(\tilde\ell(1/\theta))^2}\,d\theta=
\int_y^\infty \frac{\ell(x)}{x(\tilde\ell(x))^2}\,dx\sim
\frac{1}{\tilde\ell(y)}$.

Note that $-(\tilde\ell(x))^{-1}$ is an antiderivative of $\ell(x)x^{-1}(\tilde\ell(x))^{-2}$.

Identification of Fourier coefficients
--------------------------------------

Write $z=e^{-u+i\theta}$, $u\in[0,1]$, $\theta\in[0,\pi]$. Given a function $g_\theta(x)\ge 0$ with $|g_\theta|_\infty\le C\theta^{1-\epsilon}$ for constants $C>0$, $\epsilon\in(0,1)$, such that $x\to g_\theta(x)(1-G(x))$ is decreasing for each fixed $\theta$, define $$\begin{aligned}
J_C  =\int_0^\infty e^{-ux}\cos\theta x\, g_\theta(x) (1-G(x))\,dx, 
\quad
J_S  =\int_0^\infty e^{-ux}\sin\theta x\,  g_\theta(x)(1-G(x))\,dx.\end{aligned}$$ Let $I_C$ and $I_S$ be the corresponding integrals in the case $g_\theta\equiv1$.

\[prop-IJ\] As $u,\theta\to0^+$, $$\begin{aligned}
{2}
|I_S| & \ll \theta u^{-1}\ell(1/u), &
\quad & I_C=\tilde\ell(1/u)(1+o(1))+O(\theta u^{-1}\ell(1/u)), \\
|I_S| & \ll \ell(1/\theta), &  \quad &
I_C=\tilde\ell(1/\theta)(1+o(1))+O(u\theta^{-1}\ell(1/\theta)), \\
|J_S| & \ll \theta^{2-\epsilon} u^{-1}\ell(1/u),
& \quad &  |J_C| \ll 
\theta^{2-\epsilon} u^{-1}\ell(1/u)+\theta^{1-\epsilon}\tilde\ell(1/u), \\
|J_S| & \ll \theta^{1-2\epsilon}, &  \quad &
|J_C| \ll \theta^{1-2\epsilon}+u\theta^{-2\epsilon}.\end{aligned}$$

First, $$\begin{aligned}
|J_S| & \ll \theta |g_\theta|_\infty \int_0^\infty e^{-ux}\Bigl|\frac{\sin\theta x}{\theta 