Beurling's generalised primes

[Pieter Moree]

On Wed, 17 Dec 2003, Jon Awbrey wrote:

> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>
> marc,
>
> there's an old book by knopfmacher that discusses "generalized
> primes".

   Are these the same as the generalized primes studied extensively by
Beurling?  These were a major contribution to the theory of distribution
of prime numbers.

    John Conway

____________

Essentially, YES.

Beurling's generalised primes are norms of
generalised primes in Knopfmacher's sense.

For example, cyclic groups of prime order are examples of
generalised primes. Their norms, for which we can
take cardinality, can be considered
as Beurling's generalised primes. In this case these are
prime powers of course.

A question analogous to: how many integers are there <=x is
then: how many non-isomorphic abelian groups of order <=x are
there ?

For a definition of Beurling's generalised primes see the
review of the following paper on MathSciNet:

Diamond, Harold G.
When do Beurling generalized integers have a density?
J. Reine Angew. Math. 295 (1977), 22--39.

Let G be a free commuative semigroup with identity element 1, having a
finite or countably infinite subset P of generators (thus every element
has an unique factorisation into distinct elements of P).
G will be called a free arithmetical semigroup if in addition there
exists a homomorphism of G onto some multiplicative semigroup H of real
numbers such that for every x>0, G contains only finitely many elements
n with |n|<=x, |.| denoting the homomorphism.

The images |p| of the generators p in P
(the generalised primes) are Beurling's generalised primes,
where Beurling required on top that the sequence of numbers |p| be
unbounded. Multiplicities are allowed.

It seems that around the 1980's people stopped using this terminology. I
guess the terminology of Knopfmacher was taken over. Presumably
his book was the first in this area. It made it into the Dover series.

People are still actively doing research in this
area, for example Wen-Bin Zhang is quite active on this topic.

I wrote one chapter of my PhD thesis back in 1993 involving
generalised primes (and never
returned to them), so I am far from an expert on Beurling's
generalised primes.

Pieter Moree