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Mathematics of Computation

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Computing prime harmonic sums

Authors: Eric Bach, Dominic Klyve and Jonathan P. Sorenson
Journal: Math. Comp. 78 (2009), 2283-2305
MSC (2000): Primary 11Y16; Secondary 11Y35, 11N05, 68Q25
Published electronically: April 3, 2009
MathSciNet review: 2521290
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Abstract: We discuss a method for computing $ \sum_{p \le x} 1/p$, using time about $ x^{2/3}$ and space about $ x^{1/3}$. It is based on the Meissel-Lehmer algorithm for computing the prime-counting function $ \pi(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 4.

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Additional Information

Eric Bach
Affiliation: Computer Sciences Department, University of Wisconsin-Madison, 1210 W. Dayton Street, Madison, Wisconsin 53706

Dominic Klyve
Affiliation: Department of Mathematics, Carthage College, 2001 Alford Drive, Kenosha, Wisconsin 53140

Jonathan P. Sorenson
Affiliation: Computer Science and Software Engineering, Butler University, Indianapolis, Indiana 46208

Received by editor(s): June 19, 2008
Received by editor(s) in revised form: November 28, 2008
Published electronically: April 3, 2009
Additional Notes: E. Bach was supported by NSF grants CCR-0523680, CCF-0635355, and a Vilas Associate Award from the Wisconsin Alumni Research Foundation.
D. Klyve was supported by NSF grant DMS-0401422
J. P. Sorenson was supported by a grant from the Holcomb Awards Committee
A preliminary version of this work was presented as a poster at ANTS-VII in Berlin, Germany, July 2006.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.