We discuss a method for computing Σ 𝑝≤𝑥 1/𝑝, using time about 𝑥2/3 and space about 𝑥1/3. It is based on the Meissel-Lehmer algorithm for computing the prime-counting function 𝜋(𝑥), which was adapted and improved by Lagarias, Miller, and Odlyzko. We used this algorithm to determine the first point at which the prime harmonic sum first crosses.
Bach, Eric; Klyve, Dominic; and Sorenson, Jonathan P., "Computing Prime Harmonic Sums" (2009). Scholarship and Professional Work - LAS. Paper 98.