A Reverse Problem on Arithmetic Functions

AbstractThis paper is a response to a reverse problem on arithmetic functions of Kátai. The main result is as follows: LetBbe a fixed positive integer. On (n, B)=1,f(n) is a complex-valued multiplicative function for which |f(n)|≡1,f(n+B)−f(n)→0,n→∞. Then there must be a suitable real constantτsuch thatf(n)=niτχB(n), whereχB(n) is a Dirichlet character modB