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I know that Beurling developed a notion of generalized primes (and integers.

However, does anyone know if Beurling, or anyone else, studied subclasses of the broader class of Beurling primes that are generated with specific rules?

For example, I have in mind a method of constructing generalized primes by starting with the natural numbers N (well, N+ really), then selecting an operation/function that is not just the usual multiplication, generating a set of generalized composites (C) by operating this function on all binary combos from N, and then taking the generalized primes to be P=N-C.

My thought is that if you could look at sequences of operations (generalizing multiplication for classical primes) that generate different sequences of primes and study how the properties of those sequences change as you "move away or towards" the primes (if you had some suitable way of parameterizing the sequences of operations).

My hunch is one or more of the following is true:

(1) Buerling's studies of generalized primes (or extensions of it) already treat such constructions, either implicitly or explicitly and so I can go learn about them in his works.

(2) These alternately constructed primes may bear little to no (useful) relation to the classical primes, or may not change in any "regular" way as you move towards the classical primes that would allow one to gain information about the classical primes.

(3) Most (or all?) alternately constructed primes may not be different enough from the primes, so that while specific scales of gaps between primes, etc. may change, their overall behavior may be rather similar, so again, not very useful for finding new information about the classical primes.

Does anyone know if (1), (2) or (3) are true, and/or have any guidance about specific topics to explore in relation to looking at specific collections/sequences of generalized primes?

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I've wondered about similar questions myself. Digging through the literature, I have not been able to find treatments of such a kind.

I suspect that this is because:

  1. Most people who study the Beurling primes come from an analytical perspective, and they are mostly interested in deriving asymptotic estimates.

  2. There has been a tendency to study Beurling primes in the general case where multiplicity of generalized integers is allowed (ie. when there isn't a requirement for uniqueness of prime factoriztion).

In my opinion, it's unfortunate that a study of Beurling primes with uniqueness of prime factoriztion has been neglected, because that requirement leads to so many "nice" properties.

Specifically, the multiplication function becomes well defined. There is an isomorphism between a particular class of function $\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ and equivalence classes of Beurling integers with uniqueness of prime factorization. (If you're interested you can see more at a series of articles I wrote. I only point to my own work because, as far as I know, no one has published any material on the topic, though Lapidus and Hilberdink did come close in section 4 of "Beurling Zeta Functions, Generalised Primes, and Fractal Membranes".)

One subcase of Beurling numbers with uniqueness of prime factorization was covered by Jeffrey Largarias in his paper on the Delone property, though I don't think it talks about the multiplication functions (see "Beurling Generalized Integers with the Delone Property").

As far as (1) is concerned, I would believe that work done by the likes of Beurling, Diamond, etc. cover such constructions. They derive some general results, which should cover any case implicitly.

Also, some papers explicitly cover specific examples of Beurling number systems with special properties (eg. "Normalization of Beurling generalized primes with Riemann hypothesis" by Wen-Bin Zhang gives examples of systems "close" to the classical primes and that satisfy certain properties with the Riemann Zeta function). However, as far as I know these examples are normally done by choosing real numbers to form a sequence of Beurling primes. I'm not aware of any case in the literature when a system is defined by its multiplication function.

When considering Beurling primes with uniqueness of prime factorization, I think that we'd probably be more concerned that they are too similar to the classical primes (3) than too different (2), since after all we would be requiring them to have another property that makes them similar to the classical primes.

It may be possible to define operations to define the Beurling numbers without uniqueness of prime factorization, but I personally don't like the idea because it appears very messy. If 2, 3, 10, and 15 are primes for example, do we have $2 \cdot 15=30_1$ and $3 \cdot 10=30_2$? I don't know.

My suggestions would be:

  • If you are curious about what happens when we choose the multiplication function, see my articles. I think a lot of interesting combinatorial problems come out of this investigation. There also might be some potential for an extention of probabilistic number theory to Beurling primes with uniqueness of prime factorization.

  • If you just want to study how things change as you "move away or towards" the primes, maybe just stick with traditional material of Diamond and the rest. They've done a lot of great work.

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