Wyniki wyszukiwania

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Arithmetical aspects of certain functional equations

100%

Lucht L.

Acta Arithmetica

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1997

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tom 82

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nr 3

257-277

EN

The classical system of functional equations $ 1/n ∑_{ν=0}^{n-1} F((x+ν)/n) = n^{-s} F(x)$ (n ∈ ℕ) with s ∈ ℂ, investigated for instance by Artin (1931), Yoder (1975), Kubert (1979), and Milnor (1983), is extended to $ 1/n ∑_{ν=0}^{n-1} F((x+ν)/n) = ∑_{d=1}^∞ λ_n(d)F(dx)$ (n ∈ ℕ) with complex valued sequences $λ_n$. This leads to new results on the periodic integrable and the aperiodic continuous solutions F:ℝ₊ → ℂ interrelating the theory of functional equations and the theory of arithmetic functions.

2

On finite pattern-free sets of integers

63%

Dilcher K., Lucht L.

Acta Arithmetica

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2006

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tom 121

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nr 4

313-325

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Polylogarithms and arithmetic function spaces

63%

Lucht L., Schmalmack A.

Acta Arithmetica

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2000

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tom 95

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nr 4

361-382

4

General Dirichlet series, arithmetic convolution equations and Laplace transforms

51%

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?

Ograniczanie wyników

3 Acta Arithmetica

1 Studia Mathematica

4 Lucht L.

1 Dilcher K.

1 Glöckner H.

1 Porubský Š.

1 Schmalmack A.

1 2009

1 2006

1 2000

1 1997

Arithmetical aspects of certain functional equations

On finite pattern-free sets of integers

Polylogarithms and arithmetic function spaces

General Dirichlet series, arithmetic convolution equations and Laplace transforms