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For a unique factorization domain we know that we have some the analogues of fundamental theorem of arithmetic, and can build elements by using 'building blocks'. For me the easiest examples are integers and prime numbers, polynomials on finite fields and irreducible polynomials, and gaussian integers and the corresponding prime numbers.

My question is the following: are there any other 'similar' Euclidean domains in which the density of prime numbers is better (i.e., denser) than in the case of integers.

I am looking at possible candidates that can push the $n/\ln(n)$ as this would improve some computations needed.

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    $\begingroup$ The question will necessarily run into the question of how one counts elements. There are only a few cases where there is a natural way to count them. In number fields you have an analogue of PNT called Landau's prime ideal theorem, and something similar should hold for function fields over finite fields. Over infinite fields matters look very different - e.g when measured by height, asymptotically 100% of polynomials in $\mathbb Q[x]$ are irreducible, but I could envision other ways to count which would give a different result. $\endgroup$
    – Wojowu
    Commented Jul 21, 2022 at 11:46
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    $\begingroup$ In almost all cases (excluding number fields and function fields mentioned above) there will be infinitely many elements of norm less than a given bound. Furthermore the answer may depend on the choice of the Euclidean norm. $\endgroup$
    – Wojowu
    Commented Jul 21, 2022 at 11:59
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    $\begingroup$ Did you mean unique factorization domain when you typed unit factorization domain? $\endgroup$
    – J. W. Tanner
    Commented Jul 21, 2022 at 17:15
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    $\begingroup$ You might like to examine Knopfmacher's book Abstract Analytic Number Theory, whose goal is to develop a framework for prime number theorem type results in different areas of math. My impression is that this book did not really have the kind of impact its author had hoped; I suspect many people familiar with the prime number theorem and Landau's prime ideal theorem will never have heard of it. For a review of the book online, see maa.org/press/maa-reviews/abstract-analytic-number-theory. $\endgroup$
    – KConrad
    Commented Jul 21, 2022 at 19:54
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    $\begingroup$ Perhaps it should be kept in mind that not every unique factorization domain is Euclidean. $\endgroup$
    – Gerry Myerson
    Commented Jul 21, 2022 at 23:47

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