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

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 $$\frac{a}{n}=\frac{1}{x}+\frac{1}{y}.$$ Write $$S(N;a)=\sum _{\begin{array}{c}n\le N(n,a)=1\end{array}}R(n;a).$$ Recently Jingjing Huang and R. C. Vaughan proved that for $4\le N$ and $a\le 2N$, there is an asymptotic formula $$S(N;a)=\frac{3}{{\pi }^{2}a}\prod _{p|a}\frac{p-1}{p+1}\cdot N({\log }^{2}N+c_1(a)\log N+c_0(a))+\Delta (N;a).$$ In this paper, we shall get a more explicit expression with better error term for $c_0(a)$.

References in corpus (1)

• Mean Value Theorems for Binary Egyptian Fractions