Binary Egyptian Fractions

AbstractLet A*k(n) be the number of positive integers a coprime to n such that the equation a/n=1/m1+…+1/mk admits a solution in positive integers (m1, …, mk). We prove that the sum of A*2(n) over n⩽x is both ⪢xlog3x and also ⪡xlog3x. For the corresponding sum where the a's are counted with multiplicity of the number of solutions we obtain the asymptotic formula. We also show that A*k(n)⪡nαk+ε where αk is defined recursively by α2=0 and αk=1−(1−αk−1)/(2+αk−1)