generalized primes (twisted möbius transforms)

[Jon Awbrey]

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John Conway wrote:
> 
> On Wed, 17 Dec 2003, Jon Awbrey wrote:
> 
> > o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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> > 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

john,

i looked into the knopfmacher book some 1/4 century ago,
and did not remember beurling, but i see that there are
a couple of mentions in the first set of urls i found:

AJOL: Quaestiones Mathematicae VOLUME 24 ISSUE 3 (2001)
... of John Knopfmacher to ' Analytic Number Theory' are described.
This theory investigates semigroups with countably many generators
(generalized 'primes'), with ... 
www.inasp.info/ajol/journals/qm/vol24no3abs.html - 20k - Cached - Similar pages 

[PS] ANALYTIC AND PROBABILISTIC THEORY OF ADDITIVE ARITHMETICAL ...
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... works [3] of A. Knopf- macher and J. Knopfmacher and [5] of J. Knopfmacher and R ...
are called Beurling's generalized primes and generalized integers, respectively ... 
www.wits.ac.za/science/number_theory/research/zhang.ps - Similar pages 

[PS] John Knopfmacher, [Abstract] Analytic Number Theory, and the ... 
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... John Knopfmacher to `Abstract Analytic Number Theory' are described.
This theory investigates semigroups with countably many generators (generalized `primes'), ... 
www.wits.ac.za/science/number_theory/ research/schwarz.ps - Similar pages 

[PS] Beurling Generalized Integers with the Delone Property
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... . . The set of generalized primes of any such N consists of all but. ... .
in the sense. of Knopfmacher [11], using the trivial norm jnj = n. ... 
www.research.att.com/~jcl/doc/beurling.ps - Similar pages 

[PS] Allgemeine Dirichletreihen und
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... Das Konzept. der arithmetischen Halbgruppen wurde von Knopfmacher eingef"uhrt[Kno75].
... An Stelle von (ffi) wird dann das von Knopfmacher[Kno79] eingef. ... 
www.math.tu-clausthal.de/~matbe/ material/diss_a5_druck.ps - Similar pages 

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