On the diophantine equation \(\sum ^{k}_{i=0}1/x_ i=a/n\) (Q1078227)

The main result is the following: Let \(E_{a,k}(N)\) denote the number of natural numbers \(n\leq N\) for which the equation of the title is insoluble in positive integers \(x_ i\) \((i=0,1,...,k)\). Then \[ E_{a,k}(N)\quad \ll \quad N \exp (-C(\log N)^{1-1/(k+1)}), \] where the implied constant depends on a and k. This is an improvement on the result of \textit{C. Viola} [Acta Arith. 22, 339-352 (1973; Zbl 0266.10016)] with the replacement of \(k+1\) for k. With \(k=2\), we have the result of \textit{R. C. Vaughan} [Mathematika 17, 193-198 (1970; Zbl 0219.10023)].