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The ring of arithmetical functions with unitary convolution: Divisorial and topological properties

Jan Snellman (2004)

Archivum Mathematicum

We study ( 𝒜 , + , ⊕ ) , the ring of arithmetical functions with unitary convolution, giving an isomorphism between ( 𝒜 , + , ⊕ ) and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett [NumThe] between the ring ( 𝒜 , + , · ) of arithmetical functions with Dirichlet convolution and the power series ring [ [ x 1 , x 2 , x 3 , ⋯ ] ] on countably many variables. We topologize it with respect to a natural norm, and show that all ideals are quasi-finite. Some elementary results on factorization into atoms...

Topological aspects of infinitude of primes in arithmetic progressions

František Marko, Štefan Porubský (2015)

Colloquium Mathematicae

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...

Two problems related to the non-vanishing of L ( 1 , χ )

Paolo Codecà, Roberto Dvornicich, Umberto Zannier (1998)

Journal de théorie des nombres de Bordeaux

We study two rather different problems, one arising from Diophantine geometry and one arising from Fourier analysis, which lead to very similar questions, namely to the study of the ranks of matrices with entries either zero or ( ( x y / q ) ) , ( 0 ≤ x , y < q ) , where ( ( u ) ) = u - [ u ] - 1 / 2 denotes the “centered” fractional part of x . These ranks, in turn, are closely connected with the non-vanishing of the Dirichlet L -functions at s = 1 .

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