Mean value from representation of rational number as sum of two Egyptian fractions

AbstractFor given positive integers n and a, let R(n;a) denote the number of positive integer solutions (x,y) of the Diophantine equationan=1x+1y. WriteS(N;a)=∑n⩽N(n,a)=1R(n;a). Recently Jingjing Huang and R.C. Vaughan proved that for 4⩽N and a⩽2N, there is an asymptotic formulaS(N;a)=3π2a∏p|ap−1p+1⋅N(log2N+c1(a)logN+c0(a))+Δ(N;a). In this paper, we shall get a more explicit expression with better error term for c0(a)