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

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 $c_0(a)$

On the Inverse Problem Relative to Dynamics of the w Function

In this paper we shall study the inverse problem relative to dynamics of the w function which is a special arithmetic function and shall get some results.Comment: 11 page