Why Legendre made a wrong guess about \(\pi(x)\), and how Laguerre's continued fraction for the logarithmic integral improved it. (Q1411668)

This gives an historical account of attempts to approximate \(\pi(x)\). Laguerre's continued fraction for \(li(x)\) shows that convergence to \(\pi(x)\sim x/(\ln x- 1)\) is very slow and Legendre had available calculations of \(\pi(x)\) only for \(x\) up to \(4.10^5\). Tables are given showing the discrepancies between approximations to \(\pi(x)\) by various methods and for various values of \(x\) up to \(10^{22}\). (The largest value of \(x\) for which \(\pi(x)\) is currently known is \(4.10^{22}\)).