Computing $\psi(x)$

Let $\Lambda$ denote the Von Mangoldt function and $\begin{displaystyle} \psi(x)=\sum _{n \leq x} \Lambda(n) \end{displaystyle}$. We describe an elementary method for computing isolated values of $\psi(x)$. The complexity of the algorithm is $O(x^{2/3}(\log\log x)^{1/3})$ time and $O(x^{1/3}(\log\log x)^{2/3})$ space. A table of values of $\psi(x)$ for $x$ up to $10^{15}$ is included, and some times of computation are given.