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Topological aspects of infinitude of primes in arithmetic progressions

100%

Marko F., Porubský Š.

Colloquium Mathematicum

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2015

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tom 140

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nr 2

221-237

EN

We investigate properties of coset topologies on commutative domains with an identity, in particular, the 𝓢-coprime topologies defined by Marko and Porubský (2012) and akin to the topology defined by Furstenberg (1955) in his proof of the infinitude of rational primes. We extend results about the infinitude of prime or maximal ideals related to the Dirichlet theorem on the infinitude of primes from Knopfmacher and Porubský (1997), and correct some results from that paper. Then we determine cluster points for the set of primes and sets of primes appearing in arithmetic progressions in 𝓢-coprime topologies on ℤ. Finally, we give a new proof for the infinitude of prime ideals in number fields.

2

Remarks on systems of congruence classes

100%

Chen Y.-G., Porubský Š.

Acta Arithmetica

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1995

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tom 71

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nr 1

1-10

3

General Dirichlet series, arithmetic convolution equations and Laplace transforms

81%

Glöckner H., Lucht L., Porubský Š.

Studia Mathematica

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2009

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tom 193

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nr 2

109-129

EN

In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form $a_{d}∗g^{∗d} + a_{d-1}∗g^{∗(d-1)} + ⋯ + a₁∗g + a₀ = 0$, where $a₀,...,a_{d}: ℕ → ℂ$ are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form $∑_{x∈X} f(x)e^{-sx}$ ($s ∈ ℂ^{k}$), where $X ⊆ [0,∞)^{k}$ is an additive subsemigroup. If X is discrete and a certain solvability criterion is satisfied, we determine solutions by an elementary recursive approach, adapting an idea of Fečkan [Proc. Amer. Math. Soc. 136 (2008)]. The solution of the general case leads us to a more comprehensive question: Let X be an additive subsemigroup of a pointed, closed convex cone $C ⊆ ℝ^{k}$. Can we find a complex Radon measure on X whose Laplace transform satisfies a given polynomial equation whose coefficients are Laplace transforms of such measures?

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On the probability that k generalized integers are relatively H-prime

79%

Porubský Š.

Colloquium Mathematicum

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1981

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tom 45

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nr 1

91-99

5

On m times covering systems of congruences

50%

Porubský Š.

Acta Arithmetica

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1976

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tom 29

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nr 2

159-169

6

Covering systems and generating functions

50%

Porubský Š.

Acta Arithmetica

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1974-1975

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tom 26

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nr 3

223-231

Ograniczanie wyników

3 Acta Arithmetica

2 Colloquium Mathematicum

1 Studia Mathematica

6 Porubský Š.

1 Chen Y.-G.

1 Glöckner H.

1 Lucht L.

1 Marko F.

1 2015

1 2009

1 1995

1 1981

1 1976

1 1975

Topological aspects of infinitude of primes in arithmetic progressions

Remarks on systems of congruence classes

General Dirichlet series, arithmetic convolution equations and Laplace transforms

On the probability that k generalized integers are relatively H-prime

On m times covering systems of congruences

Covering systems and generating functions