On a number theoretic series (Q1112102)

The series \[ S(x)=1+\sum_{k\geq 1}\frac{1}{k!}\sum_{n_ 1n_ 2...n_ k\leq x;\quad n_ 1,n_ 2,...,n_ k>1}\frac{1}{\log n_ 1\cdot \log n_ 2\cdot \cdot \cdot \log n_ k} \] for \(x>1\), where \(n_ 1,n_ 2,...,n_ k\) are ordinary integers is estimated by means of Vinogradov's trigonometric sum method and contour integration. Improving the result of \textit{H. G. Diamond} [Pac. J. Math. 111, 283-285 (1984; Zbl 0541.10042)] \[ S(x)=c\cdot x+O(x \exp (-(\log x)^{1/2-\epsilon})) \] the author proves \[ S(x)=c\cdot x+O(x \exp (-c_ 1(\log x)^{3/5} (\log \log \log x)^{2/5})), \] where c and \(c_ 1\) are positive constants.