Discrete universality of \(L\)-functions for new forms (Q2508717)

Let \(F\) be a normalized newform of weight \(\kappa\). In [Izv. Math. 67, No. 1, 77--90 (2003); translation from Izv. Ross. Akad. Nauk Ser. Mat. 67, No. 1, 83--98 (2003; Zbl 1112.11026)] the authors of the paper under review proved universality in the Voronin sense for the \(L\)-function \(L(s,F)\), \(s=\sigma+it\), attached to the form \(F\). The paper under review is devoted to the discrete universality of the function \(L(s,F)\). Suppose that \(h>0\) is a fixed number such that \(\exp\{\frac{2 \pi k}{h}\}\) is irrational for all integers \(k \neq 0\). Let \[ \mu_N(\cdots)=\frac{1}{N+1} \sum_{\substack{ m=0\\ \cdots}}^{N}1, \] where in place of the dots a condition satisfied by \(m\) is to be written, \(D=\{s \in \mathbb{C}:\frac{\kappa}{2}<\sigma<\frac{\kappa+1}{2}\}\), and \(K\) is a compact subset of the strip \(D\) with connected complement. Let \(f(s)\) be a continuous function, nonvanishing on \(K\), which is analytic in the interior of \(K\). Then the authors prove that, for every \(\varepsilon>0\), \[ \liminf_{N\to \infty}\mu_N\left(\sup_{s \in K}|L(s+imh,F)-f(s)|<\varepsilon\right)>0. \] A similar assertion is also true for \(L'(s,F)\), but in this case \(f(s)\) can have zeros in \(K\). This is applied to prove that the function \(L'(s+imh,F)\) has a zero in \(D\) for infinitely many \(m\).