Golomb's arithmetical semigroup topology and a semiprime sufficiency condition for Dirichlet's theorem

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.