Let \(f\) be an additive arithmetic function. Denote by \(A(x)=\sum_{p\leq x} f(p)p^{-1}\) and \(B(x)=\sum_{p\leq x}f^2(p)p^{-1},\) where \(p\) is prime. Suppose that \[ \mu_x=(\sup_{p\leq x}| f(p)|)/\sqrt{B(x)}\to 0\text{ as }x\to \infty. \] Then it is well known that as \(x\to \infty\), \[ F_x(v)=(1/x) \sum_{\substack{n\le x \\ f(n)- A(x)