Multiplicative functions with small increments

Abstract. Let \(k\in\{1,2,3\}\). For a polynomial \(P(x)=a_0+a_1x+\cdots+a_kx^k\in {\mathbb{C}}[x]\) let \(P(E)f(n)=a_0f(n)+a_1f(n+1)+\cdots+a_kf(n+k)\). We give all multiplicative functions \(f\) which satisfy the relation $$ \sum_{n\le x}{{\vert P(E)f(n)\vert}\over n} =O(\log x). $$ In the case \(P(x)=(E^B-I)^k\), we also give all completely multiplicative function with the conditions \(\vert f(n)\vert=1\) if \((n,B)=1\) and \(f(n)=0\) if \((n,B)>1\) which satisfy $$ \sum_{n\le x}{{\vert P(E)f(n)\vert}\over n} =o(\log x). $$ where \(B\) is a positive integer.