Sums over primitive sets with a fixed number of prime factors
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- by Jonathan Bayless, Paul Kinlaw and Dominic Klyve HTML | PDF
- Math. Comp. 88 (2019), 3063-3077 Request permission
Abstract:
A primitive set is one in which no element of the set divides another. Erdős conjectured that the sum \begin{equation*} f(A) := \sum _{n \in A} \frac {1}{n \log n} \end{equation*} 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.References
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Additional Information
- Jonathan Bayless
- Affiliation: Department of Mathematics, Husson University, 1 College Circle, Bangor, Maine, 04401
- MR Author ID: 769072
- Email: baylessj@husson.edu
- Paul Kinlaw
- Affiliation: Department of Mathematics, Husson University, 1 College Circle, Bangor, Maine, 04401
- MR Author ID: 902693
- Email: kinlawp@husson.edu
- Dominic Klyve
- Affiliation: Department of Mathematics, 400 E University Way, Central Washington University, Ellensburg, Washington 98926
- MR Author ID: 776121
- Email: dominic.klyve@cwu.edu
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 3063-3077
- MSC (2010): Primary 11N25, 11Y55
- DOI: https://doi.org/10.1090/mcom/3416
- MathSciNet review: 3985487