Computing Prime Harmonic Sums

Eric Bach
Dominic Klyve
Jonathan P. Sorenson, Butler University

Abstract

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.