Mean Value from Representation of Rational Number as Sum of Two Egyptian Fractions

TL;DR

This paper refines the asymptotic formula for counting solutions to a specific Egyptian fraction equation, providing a more explicit expression and improved error estimates for the constant term.

Contribution

It offers a more explicit expression and better error bounds for the constant term in the asymptotic formula for solutions to the Egyptian fraction equation.

Findings

Derived a more explicit formula for the constant term c_0(a).

Improved the error term in the asymptotic estimate.

Extended understanding of solutions to the Egyptian fraction representation.

Abstract

For given positive integers $n$ and $a$, let $R(n;\,a)$ denote the number of positive integer solutions $(x,\,y)$ of the Diophantine equation $$ {a\over n}={1\over x}+{1\over y}. $$ Write $$ S(N;\,a)=\sum_{\substack{n\leq N (n,\,a)=1}}R(n;\,a). $$ Recently Jingjing Huang and R. C. Vaughan proved that for $4\leq N$ and $a\leq 2N$, there is an asymptotic formula $$ S(N;\,a)={3\over \pi^2a}\prod_{p|a}{p-1\over p+1}\cdot N(\log^2N+c_1(a) \log N+c_0(a))+\Delta(N;\,a). $$ In this paper, we shall get a more explicit expression with better error term for $c_0(a)$.