$N$ or on $m$, $z$ or $\tau$.

We shall take $\tau=\rho_{12}$, a function of $n$, in which case, since $$\bigl|\bigl|\sin(\ell\tau\pi)\bigr|-\bigl|\sin(\ell\rho_0\pi)\bigr| \bigr|\leq\ell \pi|
\rho_{12}-\rho_0|,$$ we have: $$\label{eq2.27} \bigl|\sin(\ell\tau\pi)\bigr|+\ell\pi|\rho_{12}-
\rho_0| \geq\bigl|\sin(\ell\rho_0\pi)\bigr| =\sin \bigl(\pi\langle
\ell\rho_0\rangle \bigr) \geq2 \langle\ell\rho_0\rangle
\geq2 C_2(\delta) \ell^{-\eta-\delta}$$ for any given $\delta>0$ and all $\ell\geq1$, where $C_2(\delta)>0$ depends on $\delta$ but not on $\ell$, the last inequality in (\[eq2.27\]) follows from the assumption that $\rho_0$ is of type $\eta$, and the second-last inequality comes from the fact that $0\leq\langle x\rangle
\leq
{\frac{1}{2}}$ for all real numbers $x$, and $\sin(\pi x)\geq2 x$ whenever $0\leq x\leq{\frac{1}{2}}$. If $$\label{eq2.28} |\rho_{12}-\rho_0|\leq C_2(
\delta) \pi^{-1}m^{-(1+\eta+\delta)}$$ then it follows from (\[eq2.27\]) that $|\sin(\ell\tau\pi)|\geq
\langle
\ell
\rho_0\rangle$ for $1\leq\ell\leq m$, and so $$\label{eq2.29} \sum_{\ell=1}^m
\frac{1}{\ell|\sin(\ell\tau\pi)|} \leq\sum_{\ell=1}^m
\frac{1}{\ell\langle\ell\rho_0\rangle} .$$

A standard argument for bounding the discrepancy of a sequence (see, e.g., p. 123 of [@Kui74]) can be used to show that, since $\rho_0$ is an irrational number of type $\eta$, =6.5pt =6.5pt $$\label{eq2.30} \sum_{\ell=1}^m
\frac{1}{\ell\langle\ell\rho_0\rangle} =O \bigl(m^{\eta-1+\delta} \bigr)$$ for all $\delta>0$. Therefore, provided that (\[eq2.28\]) holds, we can deduce from (\[eq2.26\]) and (\[eq2.29\]) that $$\label{eq2.31} \chi(N,q,\rho_{12}) \leq C_3(q)
m^{-1} \bigl(N+m^{\eta+\delta} \bigr) =O \bigl(N^{1-(1/\eta)+\delta_1}
\bigr),$$ where $\delta_1>0$ and the inequality holds for all $m$ and the identity is true 