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Gaussian Integers

Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, Yasunari Shidama (2013)

Formalized Mathematics

Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

Gaussiana

František Josef Studnička (1877)

Časopis pro pěstování mathematiky a fysiky

General Dirichlet series, arithmetic convolution equations and Laplace transforms

Helge Glöckner, Lutz G. Lucht, Štefan Porubský (2009)

Studia Mathematica

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 ) + + a g + a = 0 , where a , . . . , a d : 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 - s x ( s k ), where X [ 0 , ) k is an additive subsemigroup. If X is discrete and a certain solvability criterion is satisfied,...

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