Lemma 4.2 . [03FH]

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Lemma 4.2.

If a point $x$ of $Z_{a}$ lies in $X(U_{v}^{\beta})$ , then

	$\displaystyle\|\langle v,\log\|z\|\rangle+\lambda(v)\|\leq C(\beta)e^{-\beta}$
	$\displaystyle\|\langle v,\operatorname{Arg}(z)\rangle+\theta_{v}\|\leq C(\beta)e^{-\beta},$

where $C(\beta)\to\log|\Delta_{\mathbb{Z}}|$ as $\beta\to\infty$ .