A number theoretic series of I. Kasara (Q1835947)

\textit{I. Kasara} [Tr. Samark. Gos. Univ. Im. Alishera Navoi, Nov. Ser. 235, 64--66 (1973; Zbl 0541.10041)] considered the series \[ S(x) = 1+\sum_{k\geq 1}\frac{1}{k!}\sum_{n_ 1,\ldots,n_ k\leq x,n_ i>1}\frac{1}{\log n_ 1 \dots \log n_ k}, \] and he asserted that \(S(x) = x+O(x/\log x)\). In this note the author shows that this assertion is not correct. He proves with the aid of one of his theorems concerning Beurling generalized primes that \[ S(x) = c x+O(x\,\exp (-(\log x)^{1/2-\varepsilon})) \] with a constant \(c>1\).