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2016 Golomb's arithmetical semigroup topology and a semiprime sufficiency condition for Dirichlet's theorem
Chris Orum
Rocky Mountain J. Math. 46(3): 939-949 (2016). DOI: 10.1216/RMJ-2016-46-3-939

Abstract

Dirichlet's theorem on the distribution of primes in arithmetic progressions states that every positive integer sequence $\{an+b\mid n\geq 0\}$ with $a$ and $b$ coprime contains infinitely many primes. In 1959, Golomb pointed out that, by taking such arithmetic progressions as a base for a topology $\mathcal D$ on the positive integers, the resulting topological space $(\mathbb Z^+$, $\mathcal D)$ is both Hausdorff and connected. More recently, Knopfmacher and Porubsky showed that $(\mathbb Z^+$, $\mathcal D)$ is a topological semigroup under multiplication. After revisiting this result, we show that Dirichlet's theorem is implied by the statement that the $\mathcal D$-closure of the primes contains the semiprimes.

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Chris Orum. "Golomb's arithmetical semigroup topology and a semiprime sufficiency condition for Dirichlet's theorem." Rocky Mountain J. Math. 46 (3) 939 - 949, 2016. https://doi.org/10.1216/RMJ-2016-46-3-939

Information

Published: 2016
First available in Project Euclid: 7 September 2016

zbMATH: 06628761
MathSciNet: MR3544841
Digital Object Identifier: 10.1216/RMJ-2016-46-3-939

Subjects:
Primary: 11A41, 11B25, 11N13, 54H99

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

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Vol.46 • No. 3 • 2016
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