On some pairs of multiplicative functions correlated by an equation. II (Q1577681)

Let \(k\) be a positive integer and \(f\) and \(g\) multiplicative functions satisfying the equation \(f(n)= g(n+k)\) for all natural numbers \(n\). In a previous paper [New Trends Probab. Stat. 4, 191-203 (1997; Zbl 0943.11045)] the authors showed that either the sets \(S_f:= \{n\in \mathbb{N}: f(n)\neq 0\}\) and \(S_g:= \{n\in \mathbb{N}: g(n)\neq 0\}\) are finite or \(f(n)\neq 0\) holds for all \(n\) coprime to \(k\). In the present paper the authors investigate the first case in more detail. It is proved that \(S_f= \{1\}\) or \(S_f= \{1,p^\delta\}\) with some prime power \(p^\delta\) except for \(k\) being a Mersenne prime or \(k\in \{3,8,20, 90\}\). In these cases the set \(S_f\) is exactly determined, too.