Continuity of the hyperbolic zeta function of lattices (Q1280656)

Let \(\zeta(\Lambda,s)= \sum_{x\in \Lambda\setminus \{0\}} (\prod_{i=1}^n \max\{1,| x_i|\})^{-s}\), where \(\Lambda\) is an \(n\)-dimensional lattice in \(\mathbb{R}^n\) and \(s\in \mathbb{C}\), \(\text{Re }s>1\). The authors prove that \(\zeta (\Lambda,s)\), uniformly in any half-plane \(\text{Re }s\geq \sigma_0>1\), as soon as \(\Lambda_m\to \Lambda\) in the metric space of lattices.