On multiplicative functions with shifted arguments

Abstract. It is proved that for given integers \(a>0,~c>0~), ~(b,~d\) with \(ad-cb\neq 0\) there exists a constant \(\eta>0\) with the following property: If unimodular multiplicative functions \(g_1,g_2\) satisfy \(\vert g_1(p)-1\vert < \eta\) and \(\vert g_2(p)-1\vert < \eta\) for all \(p\in{\cal P}\), then $$ \liminf_{x\to\infty}{1\over x}\sum_{ n\le x}\vert g_1(an+b)-\Gamma\, g_2(cn+d)\vert=0 $$ may hold with some \(\Gamma\in {\Bbb C}\setminus \{0\}\) if \(g_1(n)=g_2(n)=1\) for all positive integers \(n\in {\Bbb N}\), \((n,ac(ad-cb))=1\).