I have a somewhat unconventional view of the Prime Number Theorem as a "quantification" of the infinitude of primes. Here I recall the argument of Furstenberg. Define a topology $\mathcal{X}$ on $\mathbb{Z}$.

- $\varnothing \in \mathcal{X}$
- $a \, \mathbb{Z} + b \in \mathcal{X}$

By unique factorization, there is a finite set that is the complement of these open set:

$$ \mathbb{Z} \backslash \{ -1, 1 \} = \bigcup_p S(p,0) $$

This could probably work for any number field. Let's see if I write it correctly. Is it clear?

$$ K \backslash \{ \text{units} \} = \bigcup_p S(p,0) $$

There's few literature I could find on the Furstenberg Topology itself [1] Perhaps it has a generalization that is more commonly discussed? Does the Furstenberg topology have any deeper meaning that we know about?

Later in his life he should prove Szemeredi Theorem which is entirely also about arithemetic sequences.

About two years ago, I asked for infinite Gaussian primes in segments. In standard algebraic number theory these are all merged into questions about Dirichlet and Hecke characters. One person responded that we could use **sieves**. and filter out the primes that we need.

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a priorianswer the question. $\endgroup$ – Fan Zheng Sep 12 '17 at 18:34