Computing 𝜓(𝑥)

Abstract

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