Multiplicative functions with regularity properties. I (Q788755)

The author studies multiplicative functions which satisfy some kind of asymptotic recursion relation. A typical result (Theorem 2.1) is the following. Let \(B\in\mathbb{N}\) be fixed. On the set of positive integers which are coprime to \(B\) assume (a) \(f\) multiplicative, (b) \(| f(n)| =1, c\), \(f(n+B)-f(n)=O(n^{-\gamma})\) \((n\to \infty,\ \gamma>0)\) Then \(f(n)=\chi_B(n) n^{i\tau}\) for \((n,B)=1\), where \(\chi_B\) is a character mod \(B\). The proofs are elementary. For Part II see the following review Zbl 0532.10004.