On the universality theorem
Let g(m) be a multiplicative arithmetic function, \(| g(m)| \leq 1\). Suppose that for \(x\to \infty\) \[ \sum_{m\leq x}g(m)=O(x^{\delta}),\quad 0\leq \delta<1,\inf_{p}| g(p)|>0,\quad \sum^{\infty}_{r=1}| g(2^ r)| 2^{- r\alpha}<1,\quad \alpha =(2+\delta)/3. \] Let Z(s) denote the series \(\sum^{\infty}_{m=1}g(m)m^{-s}\), Re s\(>1\), having analytic continuation. A version of the universality theorem due to \textit{B. Bagchi} (Thesis 1981) is proved for the function Z(s): Let K be a compact, simply connected and locally path connected subset of the strip \(\alpha<Re s<1\). Let f(s) be any continuous function on K which is non- vanishing on K and analytic in the interior of K. Let \(\epsilon>0\). Then \[ \lim_{\overline{T\to \infty}}(1/2T) mes\{\tau \in [- T,T],\sup_{s\in K}| Z(s+i\tau)-f(s)|<\epsilon \}>0. \]
scientific article; zbMATH DE number 3853196