The author gives remainder term estimates for the distribution function \[ F_n(x) = {1 \over n} \cdot \# \Biggl\{m \leq n,\;h_n(m) - \sum_{p \leq n} {h_n(p) \over p} {1 \over s}, \] if \( \beta_{ns} \to 0 \) for some \( s > 2 \), where \[ L_\alpha (F_n, \Phi) = \left( \int^\infty_{-\infty} \left|F_n(x) - \Phi (x) \right|^\alpha dx \right)^{1 \over \alpha}. \] If \( \beta_{n3} \to 0 \), then \[ F_n(x) = \Phi(x) + {x \over 2 \sqrt{2 \pi}} \cdot e^{-{1 \over 2}x^2} \cdot \sum_{p,q\leq n \atop p \cdot q > n} {1 \over pq} \cdot h_n(p) \cdot h_n(q) + O \left( {\beta_{n3} \cdot \log {1 \over \beta_{n3}} \over 1 + x^2} \right), \] and, if \( \beta_{n3} \to 0 \) (as \( n \to \infty \)), \[ L_\alpha (F_n, \Phi) \ll \beta_{n3} \cdot \log {1 \over \beta_{n3}} . \]