The universality theorem for \(L\)-functions associated with ideal class characters (Q2717637)

Let \(K\) be a finite extension of \(\mathbb{Q}\), \(O_K\) the integer ring of \(K\) and \(f\) an ideal of \(O_K\). Let \(\chi\) be a character of an ideal class group modulo \(\tilde f\). For \(\Re s>1\), the \(L\)-function treated in this paper is defined as NEWLINE\[NEWLINEL(s,\chi)=\sum_I {\chi (I)\over N(I)^s}NEWLINE\]NEWLINE where \(I\) runs through all ideals of \(O_K\) except 0 and \(N(I)\) denotes the norm of \(I\). The main result of the paper is the following NEWLINENEWLINENEWLINETheorem. Let \(n=[K:\mathbb{Q}]\). Set NEWLINE\[NEWLINE\sigma_K=\begin{cases} 1/2\qquad & \text{if }K=\mathbb{Q} \\ 1-1/n &\text{otherwise.}\end{cases}NEWLINE\]NEWLINE Let \(D=D_r(s_0)=\{s\in\mathbb{C} ||s-s_0|\leq r\}\) be a disk contained in the strip \(\sigma_K <\Re s <1\) and \(f(s)\) be a continuous function on \(D\) such that \(f(s)\neq 0\) on \(D\) and \(f(s)\) is holomorphic in the interior of \(D\). Then for every \(\varepsilon >0\), we have NEWLINE\[NEWLINE\liminf_{T\to\infty}{1\over T} \text{meas}\{\tau\in[0,T] |\max_{s\in D}|L(s+i\tau, \chi)-f(s)|<\varepsilon\}>0.NEWLINE\]NEWLINE The proof follows S. M. Voronin's method [see \S 7 of \textit{A. A. Karatsuba} and \textit{S. M. Voronin}, The Riemann zeta-function, de Gruyter (1992; Zbl 0756.11022)], and class field theory is also used.