Abstract. It is proved that if $f,g\colon{\msbm N}\to\{0,1\} $ are completely multiplicative functions such that $g(an+b)=f(n)$ is satisfied for some positive coprime integers $a$, $b$ and for every positive integer $n$, furthermore $g(p)=0$ for all primes $p| b$, then $f(n)=1$ iff $(n,b)=1$ and $g(n)=1$ for $(n,ab)=1$, $g(n)=0$ for $(n,b)>1$.

AMS Subject Classification (1991): 11N64, 11N69

Received November 18, 1996 and in revised form November 5, 1997. (Registered under 2644/2009.)