Joint universality theorem of Selberg zeta functions for principal congruence subgroups (Q2039515)

Let \(Z_j(s)\) \((j=1,\ldots,r)\) be the Selberg zeta function of the principal congruence subgroup of \(\mathrm{PSL}(2,\mathbb Z)\) of level \(N_j\). Assume that \(N_j\) are coprime. Let \(K_j\) be a compact subset of the strip \(\frac{\alpha+1}2<\Re(s)<1\), where \(\alpha\) is the exponent in the error term of the prime geodesic theorem for all congruence subgroups \(\Gamma\): \[ \pi_\Gamma(x)=\mathrm{Li}(x)+O(x^\alpha). \] The author proves for any non-vanishing and continuous function \(f_j(s)\) on \(K_j\) that \[ \liminf_{T\to\infty}\nu_T\left(\max_{0\ge r\ge r}\max_{s\in K_j}|Z_j(s+i\tau)-f_j(s)|<\varepsilon\right)>0 \quad(\text{ for all } \varepsilon>0), \] where \(\nu_T(\cdots)=\frac1T\mu\{\tau\in[0,T]:\cdots\}\) with \(\mu\) the Lebesgue measure on \(\mathbb R\). This is a generalization of the theorem of [\textit{P. Drungilas} et al., Forum Math. 25, No. 3, 533--564 (2013; Zbl 1328.11093)], where they proved the case of \(r=1\) and \(N_1=1\).