On a certain class of infinite products with an application to arithmetical semigroups

Indlekofer, Karl-Heinz; Manstavičius, Eugenijus; Warlimont, Richard

doi:10.1007/bf01200088

Cited by 35 publications

(24 citation statements)

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“…(n) depends strongly on the presence of a zero and the parity of k. This phenomenon can easily be explained through the prime element theorem for arithmetical semigroups (G, ~) with axiom A # according to which the prime elements p with ~(p) odd prevail in case of the presence of a zero ( [2], Corollary to Theorem 1).…”

Section: ~6mentioning

confidence: 96%

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Arithmetical semigroups III: Elements with prime factors in residue classes

Warlimont

1995

Monatshefte f�r Mathematik

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Abstract.We study the distribution of elements in an additive arithmetical semigroup (G, 6) (as introduced by John Knopfmacher) in whose canonical decomposition the degrees of the prime elements belong to a given union of residue classes mod k. Uniformity cannot be expected, a fact which is exhibited by the example K(1; 4) < K(3; 4).In the present article, we are going to solve the analogous problem in arithmetical semigroups (G, 8) with axiom A # ([3], p. 7). For the reader's convenience we repeat what it means: G is a commutative semigroup with identity e. It contains a subset P (the set of prime elements p of G) which is such that every element -r e of G can be written uniquely apart from the order of factors as a product of elements of P. ~ is a mapping G ~ ~0 such that (1) ~3(ab) = 8(a) + 8(b) for all a,b~G.(2) 7(n):= #{9~GIc~(9) = n} < oo for all n~No.This implies 8(a)=0 iff a =e. In particular, 7(0)= 1 and G is countable.1991 Mathematics Subject Classification: 11N45 (11N25).

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Section: ~6mentioning

confidence: 96%

“…In [2] it was shown that Z G has at most one zero on I z[ ~< q-i. If it exists then it is to be found at -q-1 and simple.…”

Section: ~6mentioning

confidence: 98%

Arithmetical semigroups III: Elements with prime factors in residue classes

Warlimont

1995

Monatshefte f�r Mathematik

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“…There are examples (see Indlekofer-Manstavicius-Warlimont [4] and Remark 1) of additive arithmetical semigroups G satisfying (1) with v 1/2 and # -1 = Z c(-q ) = 0. On the other hand Theorem 1 (see [4]) shows that, if G satisfies (1) with v < 1/2,…”

Section: Axiom a # There Exist Constants A > O Q > 1 Ands With O <mentioning

confidence: 97%

“…This paper will give a report on recent results about such a prime number theorem ( [1], [4]) and concerns itself with the intrinsic connection between the prime number theorem and a (new) Axiom A-#.…”

Section: Introductionmentioning

confidence: 96%

The abstract prime number theorem for function fields

Indlekofer

1993

Acta Math Hung

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“…(ii) An additive arithmetical semigroup where the "non-classical" abstract prime number theorem first examined by Indlekofer et al [9] (cf. also Knopfmacher and Zhang [11, section 5]), …”

Section: Definition 2·8 An Additive Number Systemmentioning

confidence: 99%

Asymptotic density in quasi-logarithmic additive number systems

Nietlispach

2008

Math. Proc. Camb. Phil. Soc.

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We show that in quasi-logarithmic additive number systems A all partition sets have asymptotic density, and we obtain a corresponding monadic second-order limit law for adequate classes of relational structures. Our conditions on the local counting function p(n) of the set of irreducible elements of A allow situations which are not covered by the density theorems of Compton [6] and Woods [15]. We also give conditions on p(n) which are sufficient to show the assumptions of Compton's result are satisfied, but which are not necessarily implied by those of Bell and Burris [2], Granovsky and Stark [8] or Stark [14].

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On a certain class of infinite products with an application to arithmetical semigroups

Cited by 35 publications

References 1 publication

Arithmetical semigroups III: Elements with prime factors in residue classes

Arithmetical semigroups III: Elements with prime factors in residue classes

The abstract prime number theorem for function fields

Asymptotic density in quasi-logarithmic additive number systems

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