On a constant related to the prime counting function (Q327327)

Recall that \(\pi(x)\) denotes the number of primes not exceeding \(x\), and let \[ S(x)=\sum_{2\leq n\leq x}\frac{1}{\pi(n)}-\left(\frac{1}{2}\log^2 x-\log x-\log\log x\right). \] It is known that \(S(x)\) tends to a limit, say \(C\), as \(x\to\infty\). In the paper under review, the authors prove that \(6.6840<C<6.7830\). More precisely, they obtain some explicit bounds concerning \(S(x)\).