Universality of Hecke \(L\)-functions in the Grossencharacter-aspect. (English) Zbl 1041.11060

Let \(K=\mathbb{Q}(i)\). For an ideal \(\mathfrak a=(\alpha)\in K\), the \(m\)th Grössencharacter is given by \(\lambda^m(\mathfrak a):=(\alpha/| \alpha| )^{4m}\) for \(m\in\mathbb Z.\) For \(\operatorname{Re} s>1\), the Hecke \(L\)-function is defined by \[ L(s,\lambda^m)= \sum_{\mathfrak {a}} {\lambda^m (\mathfrak a)\over N(\mathfrak a)^s}. \] Let \(C\) be a compact subset in the strip \(1/2 <\operatorname{Re} s <1\). Let \(f(s)\) be any continuous function on \(C\) such that \(f(s)\neq0\) on \(C\) and \(f(s)\) is holomorphic in the interior of \(C\). Then the authors prove that for every \(\varepsilon >0\), \[ \liminf_{T\to \infty}{1\over T^2} \mu\biggl(\bigl\{(t, m)\in[0,T] \times\{0,\dots, [T]\}\,| \, \max_{s\in C}| L(s+it, \lambda^m)-f(s)| < \varepsilon\bigr\}\biggr)>0, \] where \(\mu\) is the product measure on \(\mathbb{R}\times\mathbb{Z}\). Universality theorems for Hecke \(L\)-functions in \(t\)-aspect was proved by H. Mishou [Acta Arith. 98, No. 4, 395–410 (2001; Zbl 0982.11050), ibid. 110, No. 1, 45–71 (2003; Zbl 1031.11052)]. The \(\chi\)-universality of the Dirichlet L-function was investigated by S. M. Gonek [Ph. D. Thesis, University of Michigan (1979)], by B. Bagchi [Ph. D. Thesis, Indian Statistical Institute, Calcutta (1981)], and by K. M. Èminyan [Math. Notes 47, No. 6, 618–622 (1990); translation from Mat. Zametki 47, No. 6, 132–137 (1990; Zbl 0713.11058)].


11M41 Other Dirichlet series and zeta functions
11R42 Zeta functions and \(L\)-functions of number fields
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