  Let $a, b,c $ and $k$ be positive integers such that $1\leq a\leq
b,a<c<2(a+b), c\ne b$ and $(a,b,c)=1$. Define the arithmetic function
$f_k(a,b;c;n)$ by $$ \sum_{n=1}^{\infty}\frac{f_k(a,b;c;n)}{n^s}=\frac{\zeta
(as)\zeta (bs)}{\zeta^k(cs)}, \Re s >1.$$
  Let $\Delta_k(a,b;c;x)$ denote the error term of the summatory function of
the function $f_k(a,b;c;n).$ IN this paper we shall give two expressions of
$\Delta_k(a,b;c;x)$. As applications, we study the so-called $(l,r)$-integers,
the generalized square-full integers, the $e-r$-free integers, the divisor
problem over $r$-free integers, the $e$-square-free integers.
  An important tool is a generalization of a method of H. L. Montgomery and R.
C. Vaughan.
