The universality theorem. II (Q1057313)

The multiplicative function g(m), \(| g(m)| \leq 1\), belongs to the class \({\mathfrak M}_ M(\delta)\), if \[ \sum_{m\leq x}g(m)=x M(g)+O(x^{\delta}),\quad 0\leq \delta <1\quad for\quad x\to \infty, \] and also \(\inf_{p, g(p)\neq 0}| g(p)| \geq \eta >0\) and \(\sum^{\infty}_{r=1}| g(2^ r)| 2^{-\alpha r}<1\), \(\alpha =(2+\delta)/3\). Let Z(s) denote the analytic continuation of the Dirichlet series \(\sum^{\infty}_{m=1}g(m) m^{-s}\), where g(m)\(\in {\mathfrak M}_ M(\delta)\). In the paper universality theorems (continuous and discrete versions) of the type of \textit{B. Bagchi} (Thesis) [see also Math. Z. 181, 319--334 (1982; Zbl 0479.10028)] and some consequences for the function g(m) are obtained. [Part I, cf. ibid. 23, 283--289 (1983); translation from Litov. Mat. Sb. 23, No. 3, 53--62 (1983; Zbl 0537.10025)].