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

TL;DR

This paper extends Montgomery and Vaughan's method to derive conditional results for a class of arithmetic functions defined via zeta functions, providing new expressions for error terms and applications to various number theoretic problems.

Contribution

It introduces two new formulas for the error term of a class of arithmetic functions and applies these to problems involving special integer sets, generalizing Montgomery and Vaughan's approach.

Findings

Derived two expressions for the error term 4444

Applied results to 4444-integers and related problems

Extended Montgomery and Vaughan's method to broader contexts

Abstract

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.