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Mathematics of Computation

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Sums over primitive sets with a fixed number of prime factors


Authors: Jonathan Bayless, Paul Kinlaw and Dominic Klyve
Journal: Math. Comp. 88 (2019), 3063-3077
MSC (2010): Primary 11N25, 11Y55
DOI: https://doi.org/10.1090/mcom/3416
Published electronically: March 5, 2019
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Abstract: A primitive set is one in which no element of the set divides another. Erdős conjectured that the sum

$\displaystyle f(A) := \sum _{n \in A} \frac {1}{n \log n}$    

taken over any primitive set $ A$ would be greatest when $ A$ is the set of primes. More recently, Banks and Martin have generalized this conjecture to claim that, if we let $ \mathbb{N}_k$ represent the set of integers with precisely $ k$ prime factors (counted with multiplicity), then we have $ f(\mathbb{N}_1) > f(\mathbb{N}_2) > f(\mathbb{N}_3) > \cdots $. The first of these inequalities was established by Zhang; we establish the second. Our methods involve explicit bounds on the density of integers with precisely $ k$ prime factors. In particular, we establish an explicit version of the Hardy-Ramanujan theorem on the density of integers with $ k$ prime factors.

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Additional Information

Jonathan Bayless
Affiliation: Department of Mathematics, Husson University, 1 College Circle, Bangor, Maine, 04401
Email: baylessj@husson.edu

Paul Kinlaw
Affiliation: Department of Mathematics, Husson University, 1 College Circle, Bangor, Maine, 04401
Email: kinlawp@husson.edu

Dominic Klyve
Affiliation: Department of Mathematics, 400 E University Way, Central Washington University, Ellensburg, Washington 98926
Email: dominic.klyve@cwu.edu

DOI: https://doi.org/10.1090/mcom/3416
Keywords: Primitive sets, almost primes
Received by editor(s): February 27, 2016
Received by editor(s) in revised form: November 7, 2018, and November 25, 2018
Published electronically: March 5, 2019
Article copyright: © Copyright 2019 American Mathematical Society