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

Abstract

Let $a,b,c$ and $k$ be positive integers such that $1\le a\le b,a<c<2(a+b),c\ne b$ 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.

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• Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic
• On certain arithmetic functions involving exponential divisors
• Exponential unitary divisors