Mathematics of Computation
We discuss a method for computing Σ �≤� 1/�, using time about �2/3 and space about �1/3. It is based on the Meissel-Lehmer algorithm for computing the prime-counting function �(�), 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.
Link leads to full text made available by American Mathematical Society.
E. Bach, D. Klyve, and J. Sorenson, Computing Prime Harmonic Sums, Mathematics of Computation 78:2283-2305, 2009.