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Conditional Results for a Class of Arithmetic Functions: a variant of H. L. Montgomery and R. C. Vaughan's method

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 $$ \sum_{n=1}^{\infty}\frac{f_k(a,b;c;n)}{n^s}=\frac{ζ(as)ζ(bs)}{ζ^k(cs)}, \Re s >1.$$ Let $Δ_k(a,b;c;x)$ 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 $Δ_k(a,b;c;x)$. 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.